## Forgot Your Master Lock Combination? Learn to Crack Combination Locks

A Master Lock brand dial combination lock is one of the most popular padlocks for securing sheds, gym lockers, tool boxes, and anywhere else people store valuables. The lock's dial has 40 numbers from 0 to 39, and a combination consists of a...

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### Odds of Winning All or Nothing Lottery

All or Nothing is a simple lottery game that feels like a cross between keno and a traditional number draw game. To play All or Nothing you select 12 numbers between 1 and 24 and see how well your set matches the...

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### Cubic Equation Word Problems

Cubic polynomials are of the form ax^3 + bx^2 + cx + d where a, b, c, and d are coefficients. If the coefficients are real numbers (as opposed to imaginary or complex numbers) a cubic equation ax^3 + bx^2 + cx + d = 0...

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### Best Brain Training Card Games for Children

9 card games that teach strategy, logic, and math skills. Help your kids improve their mental acuity with these educational math and strategy card games.

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### Quadratic Equation Word Problems

In high school algebra classes, quadratic equations problems are usually presented in the form "Solve x^2 + 3x + 4 = 0" without much context or indication of how this could ever be useful in real life....

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### High School Geometry Word Problems with Solutions

Geometry and trigonometry problems are usually presented with diagrams to make it easier to visualize spatial relations. More challenging geometry problems require you to figure out the picture for yourself. Here are...

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### How to Integrate sqrt(x^3 + 1)

Graph of y = sqrt(x^3 + 1) The function f(x) = sqrt(x^3 + 1) is an example of a function with no closed-form antiderivative, in other words, you cannot work out the antiderivative in terms of elementary functions such...

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### Maximize Area with Two Subdivisions (Non-Rectangular)

Calculus Optimization: How to maximize an enclosed area with a given length of fence and two subdivisions. Solutions for non-rectangular enclosures such as circles and ovals.

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### Solving Combined Work Algebra Problems | 7 Examples

How to solve combined work math problems, combined rates in algebra, word problems with people working together.

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### 11 Banned Math Word Problems

Offensive math word problems, racist math problems, sexist word problems, math word problems about illegal activities and violence.

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### Hints and Solutions to 5 Classic Math and Logic Puzzles

The first part of this article contains short hints to help you solve the five brain teasers in the original article 5 Classic Math and Logic Puzzles. The second part contains the full solutions. Don't scroll past the...

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### Max Area of Rectangle with 3 Sections and Fixed Fence Length

In the intersection of geometry, calculus, and optimization, a common problem is to find the maximum area of a region with a fixed perimeter. In the case of a region that consists of a single section where you can't use...

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### 5 Classic Math and Logic Puzzles

Here are five classic logical brain teaser puzzles that require some mathematical knowledge, but a lot more logical reasoning skill. Some of these you may have seen before in simpler forms, so if you know how to solve...

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### How to Solve Math Problems with Guess-and-Check (Trial-and-Error)

Guess-and-check, aka trial-and-error, is a rudimentary though ancient technique for solving a variety of math problems, both simple and complex. When an equation cannot be solved with a formula (or it can but you don't...

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### Toilet Paper Math Puzzles

Toilet paper math word problems for various grades, kids and adults. Practice logical problems solving, algebra, and geometry with toilet paper.

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### Fun Facts About the Number 420

Like the number 666, the number 420 is significant in Western culture. Although it is most commonly associated with cannabis culture, the number 420 also has many curious mathematical properties. Here are some fun facts...

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### Fun Facts About the Number 210

Two Hundred Ten (210) The number two hundred ten might not seem so remarkable at first glance, but this integer is surprisingly rich in mathematical properties. It also has much significance in pop culture, art,...

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### How to Win More in a Lottery Pool

How to increase your lottery pool's chances of winning, tips and tricks for lotto pools that will give yours an edge over other players and clubs.

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### Fun Facts About the Number 17

Seventeen is a special number that appears numerous times in the Old Testament of the Bible, in many legends in Islamic traditions, and is often cited as being the "least random number" -- owing to the fact...

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### How to Calculate the Coefficient of Relatedness

Full sisters have a coefficient of relatedness equal to 1/2. The coefficient of relatedness or coefficient of relationship (CR) is how related you are to a particular family member. The closer the number is to 1, the...

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### How to Integrate Tan(x)^4 and Tan(x)^5 -- Tan^4(x) and Tan^5(x)

The fourth and fifth powers of the tangent function can be written either as tan(x)^4 and tan(x)^5, or as tan^4(x) and tan^5(x). To integrate these functions, you need to apply trigonometric identities and use either...

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### UltraLotto 6/58 Probabilities | Odds and Prizes

UltraLotto is the largest jackpot game currently offered in the Philippines. It was introduced in February 2015 and is a 6/58 game, meaning players choose 6 distinct numbers from between 1 and 58. If they match all 6...

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### Odds and Prizes for Mega-Sena Lottery

Mega-Sena is the national lottery of Brazil and is operated by the Caixa Econômica Federal Bank. It's the largest lottery in Brazil and tempts players with huge prizes. The structure of the game is a little...

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### Illinois Lotto with Extra Shot: Odds and Prizes

Lotto with Extra Shot, known simply as "Lotto" in Illinois, is a state lottery game with a jackpot prize and several smaller prize levels. The game is similar in structure to the national lottery games...

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### How to Draw a Butt on a Graphing Calculator

Equations to draw a butt on a graphing calculator, polar coordinates and implicit functions in rectangular coordinates for butt-shaped curves

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### 5 Best Math Dice Games

math dice games for teaching mathematics and critical thinking, teacher resources, classroom games with math manipulatives, easy games for kids and adults, dice games of skill and chance

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### Approximate Integral of sqrt(1 + x^4)

For positive integer values of m and n, functions of the form f(x) = (x^m + 1)^(1/n) don't have elementary antiderivatives unless m and n equal 1 or 2. In particular, the function sqrt(1 + x^4) cannot be integrated...

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### Lottery Probability for Dummies

Do you consider yourself a math dummy but still want to understand how lottery odds work? With a little patience and a willingness to learn, even dummies can become more aware of what lottery probabilities mean and even...

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### Surface Area and Volume of an Octahedron | Formula and Examples

A regular octahedron is one of the five Platonic solids. As a regular polyhedron, all of its faces are equal, and each vertex has the same degree. On a regular octahedron there are eight equilateral triangle faces, six...

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### Volume and Surface Area of a Capsule | Formula and Examples

A capsule is a solid geometric shape best described as a cylinder with two hemispheres attached at both ends. This is a common shape for tanks and of course medicinal capsules, hence the name. If you know the overall...

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### How to Pick Winning Lottery Numbers

How a lump sum lottery payment is actually delivered to a winner. Whether you play the lottery as a hobby or as a serious financial investment, chances are you've probably spent more money than you've won over the...

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### Arithmetic Card Games with a Standard Deck

"Arithmetic" is a simple math card game that can help children practice their math skills and think on their feet, but it can also be a fun competitive game for adults. For players of all ages, it's a great...

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### How to Make a Circle Cake from a Square

How to "square the circle" or alternatively "circle the square" is a classical geometry problem about constructing a perfect circle from a perfect square (and vice versa) using only a finite number...

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### Volume of an Elliptical Cone

An elliptic cone is a cone whose base is an ellipse. If you take a flexible circle-base cone and gently squeeze the sides, you will make a shape that resembles and elliptical cone. To find the volume of such a cone, you...

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### Should you buy 10 tickets for one lottery drawing, or play the lottery 10 times?

Is it better to buy X tickets for a single lottery drawing, or buy 1 ticket for X different lottery drawings? A lotto math analysis...

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### Probability of Sharing the Powerball Jackpot

A common question among avid Powerball lottery players is what the odds are that they will have to share the jackpot if they win. And if the jackpot is likely to be split among more than one winning ticket, what the...

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### Dissection Puzzles: Ostomachion, Egg of Columbus, Hexiamonds

While tangrams are the most popular dissection puzzle, there are many other shape-making or tile puzzles like tangrams. The Ostomachion (aka Stomachion and Syntomachion) is an ancient tile puzzle sometimes attributed to...

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### Sample Calculus 1 Exam Questions

min - max problems on differential calculus exams, using first derivatives to solve calculus test questions, examples and practice test problems for Calc I students in high school or college

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### Out-of-Focus Lights Images | Bokeh Lights Photography Effects

In photography, bokeh effects or bokeh is the out-of-focus background in a photo, the part that is blurred on purpose to draw attention to the foreground. Sometimes bokeh effects are behind an in-focus object being...

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### Integrals of x*sin(x)^2, x*cos(x)^2, x*tan(x)^2

How to integrate x sin2(x), x cos2(x), and x tan2(x) ... x times sin^2, cos^2, and tan^2. Trigonometric integrals with a linear factor, trig functions squared.

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### Volume of a Square to Round Transition

How to calculate the volume of a square to round transition, round to square transition, easy volume formula and its derivation

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### Oz Lotto Odds -- Australian Lottery

Oz Lotto is one of the big national lotteries available in all states and territories of Australia. It is administered by Tattersall's lottery company and each game costs $1.10 plus the agent's commission. During the...

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### Hot Aunty Math Word Problems

Unless you're from India, you may be wondering what a hot aunty is. "Hot aunty" is Indian English slang for a hot woman, the Kim Kardashians of the subcontinent. In India, "hot aunty" is the most...

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### How to Compute the Arithmetic-Geometric Mean with JavaScript

The arithmetic-geometric mean (abbreviated AGM) is a hybrid between the arithmetic mean and geometric mean. To compute the AGM of two numbers, A and B, you first find their arithmetic and geometric means, thereby...

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### How to Integrate sqrt(x)/(x+1) and sqrt(x)/(x-1)

While it is easy to integrate functions like x*sqrt(x+1) and x/sqrt(x+1) with a simple linear substitution, it is not as straight forward to integrate the square root of x over (x+1), (x-1), or any other linear term of...

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### Calculate the Area of a Rectangle Given the Diagonal

Given only the diagonal length of a rectangle, there is no unique solution for the area or dimensions; you need one other measurement: length, perimeter, or ratio.

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### How to Integrate x*sqrt(x+1) and x/sqrt(x+1)

The integrals of x*sqrt(x+1) and x/sqrt(x+1) are easy to work out with a simple linear substitution. These functions are particular examples of functions of the form x*sqrt(ax+b) and x/sqrt(ax+b), where 'a' and 'b' are...

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### Volume of a Pentagonal Prism: Formula and Examples

A pentagonal prism is a solid with two equal size pentagonal faces parallel to each other on opposite ends of the prism, with rectangles or parallelograms connecting the pentagons. If it is a right pentagonal prism, the...

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### Triangular Prism Volume & Surface Area

A triangular prism is a solid shape with two parallel triangular faces of equal size and shape, and three parallelogram faces that connect the sides of the triangles. If you have a right triangular prism, the...

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### Rectangular to Spherical Coordinates & Vice Versa

The surface x^2 + zy^2 = 20 Knowing how to convert rectangular coordinates to spherical coordinates, and how to convert spherical coordinates to rectangular coordinates, can help you express equations of surfaces in...

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### Area of a Heptagon Formula

A regular heptagon is a polygon with seven sides of equal length whose edges meet at equal angles. The interior angles at the vertices of a regular heptagon are approximately 128.5714 degrees. This is derived from the...

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### Integrals of Ln(e^x + 1) and Ln(e^x - 1)

Ln(e^x + 1) and Ln(e^x - 1) are non-integrable functions (no closed-form antiderivatives), but their integrals can be approximated with infinite series.

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### How to Solve "Volume = Surface Area" Geometry Problems

Some 3-dimensional geometric shapes have the curious property that their surface areas and volumes are equal. For example, the diagram above shows a cylinder of radius 3 and height 3 capped by a hemisphere of radius 3....

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### How to Solve "Area = Perimeter" Geometry Problems

Some 2-dimensional geometric shapes have the curious property that their area and perimeter are equal. For example, in the diagram above, the L-shape has an area of 26 square units and a perimeter of 26 linear units. In...

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### Is It a Leap Year? How to Calculate Leap Years

Leap years are years which have an extra day in the calendar, 366 instead of the usual 365. The extra day is February 29, often called Leap Day. All other years are called common years. Why do we add an extra day to...

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### How much is 30% off, 40% off? | Doing Percentages in Your Head

Learning how to mentally calculate percentages, discounts, taxes, and tips is a useful skill, even in this day and age when we have phones and calculators to do all the math for us. With practice, in less than the...

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### How to Find the Missing Side of a Trapezoid Given the Area

It's easy to find the missing side or sides of a trapezoid given two or three sides and the trapezoid's perimeter. However, it is more difficult to find the missing side of a trapezoid if you know the area and two or...

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### How to Integrate sin^5(x) and cos^5(x) | sin(x)^5 and cos(x)^5

Graph of y = sin(x)^5 in red, graph of y = 2 + cos(x)^5 in blue The graphs above show the functions sin(x) raised to the 5th power and cos(x) raised to the 5th power. These functions can be written either as f(x) =...

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### Pythagorean Theorem Word Problems

Statue of Pythagoras in Samos, Greece The Pythagorean Theorem is one of the oldest and most useful mathematical and geometric formulas. Although it was named after the Greek mathematician Pythagoras because he gave...

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### Solution to the 3 Inscribed Circles Problem

Three equal sized circles tangent to one another are inscribed within a larger circle. What is the radius of the smaller circles or the larger circle? How to solve the 3 inscribed circles problem

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### Trapezoid Area Given 4 Sides

How to find the area of a trapezoid or trapezium from the four sides, formula for volume of a trapezoid using four side lengths

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### Challenging 4th/5th Grade Word Problems | Gifted and Talented & Home School Math Exercises

4th grade and 5th grade math word problems. Challenging and difficult word problems to test problem solving skills for teachers, tutors, home schooling parents, gifted and talented programs.

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### Mollweide's Formula Proof | Triangle Trigonometry

Mollweide's Formula for the angles and sides of a triangle Mollweide's Formula (actually, a pair of formulas) is an equation that relates all three side lengths and angles of a triangle. If a triangle has sides of...

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### How to Solve Trigonometry Problems with Tangent

In right triangle, the tangent of a non-right angle is the ratio of the opposite leg to the adjacent leg. In terms of sine and cosine, the tangent function is also defined by the equation tan(x) = sin(x)/cos(x). The...

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### How to Solve Trigonometry Problems with Cosine

The cosine of a non-right angle in a right triangle is the ratio of the adjacent leg length to the hypotenuse length. It is related to the trigonometric function sine by the equations cos(θ)^2 = 1 - sin(θ)^2...

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### How to Solve Trigonometry Problems with Sine

The sine of an angle is a ratio derived from a right triangle. If θ is a non-right angle of a right triangle, then the sine of θ, written sin(θ) is the length of the opposite leg divided by the length...

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### 7 Hard Trig Problems—Can You Solve These Challenging Geometry Problems?

Trigonometry is an area of geometry that deals with lengths and angles of triangles. And since many two-dimensional figures and three-dimensional sold shapes can be broken down into triangles, trigonometry is also used...

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### Inscribed Circles and Hexagons | 7 Hard Geometry Problems

Geometry problems involving circles inscribed in regular hexagons and regular hexagons inscribed in circles can be solved the same way as problems involving circles and equilateral triangles. The key geometric...

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### Maximize the Viewing Angle | Calculus Optimization Problem

A classic problem in differential calculus is to find the optimal distance from a raised screen so that the viewing angle is maximized. The angle formed by the bottom of the screen, eye, and top of the screen will vary...

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### How to Integrate e^(e^x)

The single-variable function f(x) = e^(e^x) is one of the simplest examples of a non-integrable function. In fact, any double-exponential function of the form f(x) = A^(B^x) is also non-integrable. A non-integrable...

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### 17 Geometric Shapes You've Never Heard of

Strange Shape Names: Obsure and unusual geometric shapes and names of shapes, 17 weird shapes you probably don't know...

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### Tangent Plane Examples | Multivariable Calculus

In two-variable calculus, the tangent line on a curve y = f(x) at a point (x0, y0) is given by the equation y - y0 = f'(x0)*(x - x0) where f'(x0) is the derivative of f(x) evaluated at x0. For an implicitly defined...

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### Antiderivative of Sqrt(Cos(x)) and Sqrt(Sin(x))

Though they look simple, the square root of cosine and square root of sine are non-integrable functions, meaning their antiderivatives cannot be expressed in closed-form in terms of elementary functions. No matter how...

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### Surface Area of a Sphere Given the Circumference

How to calculate the surface area of a sphere from the circumference instead of the radius, geometry shortcuts and formulas

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### Formula for the Area of a Quadrilateral Given Side Lengths

A quadrilateral is simply any polygon with four sides. Certain quadrilaterals have special names, such as rectangle, rhombus, square (the only shape which is both a rectangle and a rhombus), trapezoid, parallelogram,...

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### Integral of Ln(1 + x^3) and Ln(x^3 - 1)

How to integrate ln(x^3 + 1), natural logarithm of cubic polynomial. Use polynomial factorization to simplify logarithm functions and find their antiderivatives...

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### How to Integrate (e^x)/x

The antiderivative of (e^x)/x cannot be expressed in closed form in terms of elementary functions; in other words, (e^x)/x is a non-integrable function. Nonetheless, you can approximate the integral of (e^x)/x with...

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### Count from 1 to 10 in 50 Languages

How to count from 1 to 10 in different foreign languages. Number words 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Arabic, French, German, Hindi, Italian, Japanese, Mandarin, Navajo, Polish, Spanish, Swahili,Tamil

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### How to Integrate Sqrt(Tan(x)) | Hard Antiderivatives

In general, the square root of a trig function is not integrable, for example, sqrt(sin(x)) and sqrt(cos(x)) are functions whose antiderivatves cannot be expressed in closed form. However, there are some notable...

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### How to Integrate sin(ax)sin(bx) | Trig Tricks

integral of sin(mx)sin(nx) and sin(ax)sin(bx)sin(cx), trigonometric transformation of products into sums

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### Surface Area of a Trapezoidal Prism

Formula for trapezoidal prism surface area, with examples.

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### How to Integrate 1/(x^4 + 1) by Hand

Antiderivatives of rational functions become more and more complicated as the degree of the denominator polynomial increases. We have already seen in other tutorials how to find the integrals of 1/(ax^2 + bx + c),...

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### How to Integrate 1/(x^4 - 1)

Antiderivative of 1/(x^4 - 1) using partial fractions. The integral of (x^4 - 1)^(-1) converges from x = a to x = infinity and from x = -infinity to x = -a.

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### Best Combination Locks for School Lockers

Getting a locker for the first time is an exciting part of starting junior high or high school. No longer do you have to carry everything with you from class to class, between home and school. To keep your books,...

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### Derivative of Sqrt(x) with Limits

Derivative of f(x) = sqrt(x) = x^(1/2) using limit definition of derivative. Proof that f'(x) = (1/2)x^(-1/2)

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### How to Integrate Sin(Ln(x)) and Cos(Ln(x))

Logarithmic trig functions like sin(ln(x)) and cos(ln(x)) arise as solutions to differential equations of the form (x^2)y'' + (ax)y' + by = 0 which are often called Euler differential equations. The graphs of the...

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### How to Circumscribe and Inscribe a Triangle

Triangles are unique among polygons in that any triangle has both a circumscribed circle and an inscribed circle. Among quadrilaterals, pentagons, hexagons, etc., only special forms can be both circumscribed and...

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### Body Math and Fitness Formulas

When starting a new weight loss plan or workout routine, it's helpful to track your progress with different measures of fitness. The fitness indicators below use height, weight, age, heart rate, skinfold thickness, and...

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### More Geometry Puzzles: Counting Hidden Shapes

Geometric shape counting puzzles and brain teasers can help develop your analytical skills and strengthen your working memory. The trick to solving these puzzles is to work out an efficient counting strategy so that you...

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### Forgot Your Master Lock Combination? Learn to Crack Combination Locks

A Master Lock brand dial combination lock is one of the most popular padlocks for securing sheds, gym lockers, tool boxes, and anywhere else people store valuables. The lock's dial has 40 numbers from 0 to 39, and a...

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### How to Reflect a Coordinate Shape Over the X-Axis, Y-Axis, or Diagonal

Using Cartesian coordinates to plot shapes and diagrams has many advantages over other methods of graphing and drawing two-dimensional and three-dimensional objects. When you know an object by its coordinates you can...

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### Geometry Brain Teasers: Shape Counting Puzzles

Among mathematical brain teasers, shape counting puzzles are among the simplest to understand, but often the hardest to work out. They rely on visual acuity and memory as much as math and geometry skills, and therefore...

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### Product Rule Problems Worked Out | Calculus Derivative Examples

The product rule in differential calculus is one of the first techniques students learn for taking derivatives. Once you get the hang of using the product rule, taking derivatives of complicated functions is a snap. The...

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### Keno Probability: How to Increase Keno Winnings

Keno is a relatively simple lottery game that is offered as an instant lottery game in many places where they sell lotto tickets, as well as in some casinos. Since it is a game of pure chance, there is no special trick...

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### Calculus Optimization: Cone of Maximal Volume Inscribed in a Sphere

Lots of classical calculus problems involve finding the optimal dimensions of a shape inscribed in a sphere so that the shape has the largest possible volume. One of the easiest examples of such a problem is finding the...

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### How to Integrate Arcsin(x) or Sin-1(x)

Graphically, the inverse sine function looks like a small segment of the sine function reflected over the diagonal line y = x. In trigonometric terms, the function y = arcsin(x) or y = sin-1(x) means that if you input...

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### "I suck at math!" How to Change Your Attitude Toward Math

Just because you're stuck on a problem doesn't mean you're bad at math. (Pixabay) Lack of mathematical confidence is a bigger problem than lack of mathematical skill. If you're one of those people who always say they...

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### Sacred Geometry: How to Draw a Flower of Life with Only a Compass

The Flower of Life is a figure in sacred geometry that consists of 19 interlocking circles. Sacred geometry is a branch of some religious, New Age, or occult belief systems that ascribes special properties to certain...

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### How to Find the Depth of a Well by Dropping a Rock

A old trick for calculating the depth of a well or deep hole is to drop a rock into the well and measure how long it takes before you hear the rock hit the bottom. If the time between dropping the rock and hearing its...

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### Fibonacci Sequence Patterns: Ratios, Sums, Identities, and Geometric Patterns

Most people know the basic recursive formula for generating the Fibonacci numbers, F(n+2) = F(n+1) + F(n) where F(0) = 0 and F(1) = 1, as well as connections between the Fibonacci numbers and the golden ratio φ =...

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### How to Integrate x(e^x)cos(x) and x(e^x)sin(x)

The method of integration by parts can be used to find the antiderivatives of two-product functions such as x*cos(x), x*e^x, and (e^x)cos(x), but what about the triple product of a polynomial, trig function and...

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### Moon Sign Formula: Calculate Your Astrological Moon Sign by Hand

When people talk about their zodiac sign, e.g., "I'm an Aquarius," what they're usually referring to is their astrological sun sign. Your sun sign is the constellation the sun was passing through (from the...

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### Volume and Surface Area of Torus (Doughnut)

A torus is one of the basic 3-dimensional curved shapes in geometry. Inner tubes of tires, doughnuts, and rings are all examples of tori (toruses) and torus-like shapes. Like a sphere, a torus has circular...

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### Beatty Sequences: Mathematical Curiosities with Irrational Numbers and Integers

A partition of a set of numbers is when you divide it into two or more non-overlapping subsets where every element of the original set is contained in one subset. Both finite and infinite sets can be partitioned. For...

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### How to Score an NCAA March Madness Bracket

After you fill out your March Madness bracket with your friends, family, or office, the question of how to award points inevitably comes up. There's no right or wrong way to score a bracket so long as everyone in your...

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### March Madness Probabilities

Think you have a shot at filling out a perfect bracket? (Public Domain Image via Pixabay) Odds are, you're not going to win Warren Buffet's billion dollar NCAA tournament bracket challenge, but you might win the office...

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### Volume & Surface Area of an Elliptic Cylinder: Formulas and Examples

formula for the volume, lateral surface area, and total surface area of an ellipse-base cylinder, and also for a stadium-base cylinder...example problems to calculate volume and surface area

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### How to Integrate 1/(1 + sqrt(x))

Integrals of the form 1/(a + b√x) are not as simple as those of the form 1/√(a + bx), though they look very similar. Rather than apply the power rule to sqrt(x) = x^(1/2), you need to use a combination of...

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### How to Convert to Base-5

Most human civilizations settled on a base-10 or base-20 numbering system because human hands have 10 fingers, and our hands and feet have 20 digits. Considering we have five fingers on one hand, it's not such a stretch...

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### How to Integrate Sqrt(x^4 + 1) and Sqrt(9x^4 + 1)

Functions of the form f(x) = sqrt(p(x)), where p(x) is a polynomial, are only integrable when the degree of p is 0, 1, or 2, or else if p(x) = q(x)^2 for some other polynomial q(x). This means the functions sqrt(x^4 +...

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### Volume of a Trapezoidal Prism: Formula and Examples

Formulas for the volume of a trapezoidal pyramid as well as several example problems to solve, with solutions.

- 2
### How to Solve y" = f(y)g(y'), Second-Order Non-Linear Differential Equation

How to solve y" = f(y, y') in the particular case y" = f(y)g(y'), second order non-linear differential equation solutions and examples such as y" = sin(y)y' and y" = (e^y)(y')^3.

- 1
### Light Refraction and Reflection in Diamonds

A diamond engagement ring (round brilliant cut) refracting and reflecting light. (Morguefile) Diamonds and other precious gems sparkle because of the way they reflect, refract, and disperse light. When light hits a...

- 2
### How to Solve y" = f(y), Second-Order Non-Linear Differential Equations

Differential equations of the form y" = f(y), or d2y/dx2 = f(y) where y is function of x and f is a non-linear function, are examples of non-linear second-order ordinary differential equations. As this equation...

- 1
### Diamond Cut Geometry

Engagement ring with three Asscher cut diamonds. Gem cuts are three-dimensional geometric designs developed to enhance the sparkle and fire of gemstones such as diamonds, emeralds, rubies, sapphires, cubic zirconia...

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### Why the Derivative of e^x Is e^x: Proof of the Exponential Derivative

Graph of y = e^x and the tangent line at x = 0.75. The slope of this tangent line is e^0.75. The function f(x) = e^x is a calculus student's favorite because it's equal to its own derivative, and as a consequence of...

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### Why There Are Infinitely Many Prime Numbers & More Curious Facts About Primes

A 17-year cicada has emerged after spending 17 years underground. (Pixabay) A prime number is a positive integer whose only factors are 1 and itself, while a composite number has more than these two factors....

- 1
### How to Convert to Base-6

The base-6 numbering system is also known as heximal or senary, and uses only the six digits 0, 1, 2, 3, 4 and 5 to represent numbers. Heximal has many interesting properties that are analogous to features of our usual...

- 0
### How to Convert to Base-12

The base-12 numbering system is also known as duodecimal and dozenal. Because 12 is divisible by the prime factors 2 and 3, one of the advantages of the duodecimal system is that fractions over 2, 3, 4, 6, 8, 9, 12, 16,...

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### Why Does d/dx[Sin(x)] = Cos(x)? Complete Proof of the Derivative of Sin(x)

Proving that the derivative of sin(x) equals cos(x) -- using nothing but the limit definition of the derivative and basic trigonometric and geometric principles -- is rather involved and requires some clever tricks that...

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### Drawing Hearts on a Graphing Calculator

How to draw a heart on a graphing calculator, heart equations for polar coordinates and parametric coordinates...

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### Why does d/dx[Ln(x)] = 1/x? Proof That the Derivative of Ln(x) Equals 1/x

The derivative of the natural logarithm function Ln(x) is one of the first derivatives you learn in Calculus I. Unfortunately, not everyone who takes calculus learns why the derivative of Ln(x) is 1/x. If you took a...

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### How to Integrate 1/(x^3 + 1) and 1/(x^3 - 1)

Antiderivatives of 1/(x^3 + 1) and 1/(x^3 - 1) step by step using partial fractions

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### How to Integrate e^(1/x)

How to find the integral of e^(1/x) with series, and approximate integration techniques for functions with non-elementary antiderivatives.

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### Integral of y = x^x

How to integrate f(x) = x^x, does x^x have an antiderivative or is it an non-integrable function? How to estimate integrals of x^x...

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### Best Graphing Calculators Similar to the TI-84 Plus

When students need to buy a graphing calculator, they usually pick a Texas Instruments 80-something model such as the TI-84 or TI-84 Plus. However, there are other graphing calculators on the market similar to the TI-84...

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### Best TI-84 Alternatives

Choosing the best alternative to the TI-84 or TI-84 Plus graphing calculator depends on what features you desire in a calculator and your needs as a student, teacher, researcher or professional. If you want a graphing...

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### How to Integrate 1/(ax^2 + bx + c) -- Antiderivative of a Quadratic Rational Function

The function f(x) = 1/(ax^2 + bx + c) is a rational function with a quadratic denominator. There are three cases to consider for the quadratic ax^2 + bx + c with real number coefficients: the expression has two distinct...

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### How to Integrate sin(x)/x and cos(x)/x

Two simple functions, sin(x)/x and cos(x)/x, can't be integrated the easy way. Here is how it's done.

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### How to Find the Area and Arc Length of a Logarithmic Spiral

A logarithmic spiral, also called an equiangular spiral, is one whose distance from the center increases exponentially with every turn. The most convenient way to describe a logarithmic spiral mathematically is with a...

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### How to Calculate a Number Raised to a Complex Power

In high school you probably learned how to multiply, add, subtract, and divide complex numbers, but what you probably didn't learn is how to perform more advanced operations with complex numbers. For example, you likely...

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### How to Integrate sin(x^2) and cos(x^2)

The functions f(x) = sin(x^2) and g(x) = cos(x^2) are examples of non-integrable functions, that is, functions whose antiderivatives cannot be expressed in terms of elementary functions. They can be expressed in terms...

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### How to Integrate x*sin^2(x) or x*sin(x)^2

The function equal to x times the square of the sine of x can be written two different ways, either f(x) = x*sin(x)^2 or f(x) = x*sin^2(x), as shown in the image above. The choice of notation is up to personal...

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### Hanging Cable Problems | Catenary Math

Equations for catenaries given the distance between the ends of the hanging rope, the length of the rope, or the length of sag. How to solve problems with hanging cables, chains, or ropes.

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### Solving Hardy-Weinberg with 3 Alleles

Click here to go to example problems. In biology, a population of organisms of the same species is said to be in "Hardy-Weinberg equilibrium" if the allele and genotype frequencies remain stable over...

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### Physics with Calculus: Solving Work Integrals (Rope and Spring)

Integral calculus can be applied to physics problems involving work, specifically problems that involve applying a force over a certain distance. In physics, work is measured in Joules and its units are 1 J = 1 kg *...

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### Best Books to Teach Yourself Calculus

The best books to teach yourself calculus if you are creating a self-study calculus program rather than taking a college class...calculus textbooks

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### 100 Free Fractal Images Organized by Color

Generation of a simple fractal, the von Koch snowflake Fractals are a popular type of mathematical artwork generated by sets that exhibit infinite repetition and patterns. A fractal repeats itself on a smaller scale...

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### How Do You Find the Angles of a Triangle from the Three Sides?

A typical problem in geometry is to find the angles of a triangle from the lengths of the three sides. If you have the physical object, you can just use a protractor to measure the angles. But if you are are trying to...

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### Trigonometric Substitution: Solved Integral Calculus Problems

Trigonometric substitution is a clever integration technique used to find the antiderivatives of functions that involve the square root of a quadratic polynomial, i.e., a second degree polynomial of the form Px^2 + Qx +...

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### Antiderivative of Sqrt(1 + Cos(x)) and Similar Functions

Graph of y = sqrt(1 + cos(x)). Finding the integral of sqrt(1 + cos(x)), sqrt(1 - cos(x)), sqrt(1 + sin(x)), sqrt(1 - sin(x)) and similar functions may seem challenging since there is no obvious way to apply a...

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### How to Integrate Sec(x)^3 and Sec(x)Tan(x)^2 -- Trig Antiderivatives

Graph of y = sec(x)^3. Finding the antiderivatives of f(x) = sec(x)^3 and g(x) = tan(x)^2*sec(x) are challenging problems in integral calculus because standard tricks like integration by parts, substitution, and using...

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### Easy Triangle Quilting Patterns

You can create an endless number of unique quilt designs using half-square triangles. Half-square triangles, also known as isosceles right triangles, tessellate perfectly and can be arranged to create stars, flowers,...

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### Easy Rectangle Quilt Patterns

Easy rectangle quilt patterns -- pinwheels, herringbone, bricks, spirals, tri-color blocks. Simple rectangle patterns for quilting with same size blocks arranged into unique and interesting designs.

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### Penrose Tiling Quilt Patterns

A Penrose tiling -- named for its discoverer Roger Penrose -- is a non-periodic quasicrystal tiling pattern based on a regular pentagon. A non-periodic or aperiodic tiling lacks translational symmetry; a quasicrystal...

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### Rounding Error and Its Consequences

The round-off error of a number is the difference between its rounded value and its exact value. Some rounding error is unavoidable in solving real-world math problems that involve irrational numbers, for example,...

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### Best Calculation Tools for Quilting

Patchwork quilting, sewing, and other fabric crafts require a lot of calculations, such as finding the yardage of fabric needed, computing seam allowances, figuring out patchwork shapes, scaling small images to make a...

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### Best Calculators for the FE Exam -- Fundamentals of Engineering

The Fundamentals of Exam Exam (FE Exam) is administered by the National Council of Examiners for Engineering and Surveying (NCEES), and passing this standardized test is one of the first steps in earning a Professional...

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### Surface Area of the Human Body

A person can accurately calculate the total volume of his body by submerging himself in a tank of water and measuring how much water is displaced. But measuring surface area is much trickier. There is no simple physical...

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### How to Calculate the Area of a Circle from Its Circumference

The most commonly taught formula for the area of a circle uses the radius or diameter. If the radius is R and the diameter is D = 2R, then the area of the circle is given by the equations area = πR^2 area =...

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### 5 Practical Formulas Adults Should Know

5 formulas everyone should know, practical math formulas for adults

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### Math Formulas Everyone Should Know

Two giraffes discussing the Navier-Stokes equation. (Image courtesy of Pixabay) What good is it to know how to find the derivative of a cubic polynomial or how to calculate the squared sum of the cubes of the first 500...

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### Formula for the Arc Length of a Helix

helix length formula and examples, length of string wrapped around a tube, length of wire coiled around a cylinder,...

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### Truncated Cone Formulas

Truncated cone with height H and radii A and B. Everything you ever wanted to know, and things you didn't even know you wanted to know about truncated cones all in one place. When it comes to truncated cones, aka...

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### How to Calculate the Surface Area of a Dome

A dome is a shape with a flat circular bottom, rounded sides, and a rounded top. Apart from those requirements, domes can vary in steepness and curvature. Three basic varieties of dome shapes that allow you to calculate...

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### Conical Frustum / Truncated Cone: Geometry Problems with Solutions

Problems involving conical frustums (truncated cones) can be particularly tricky since they are not quite cones, and not quite cylinders, but a shape in between with characteristics of both. Whereas cones and cylinders...

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### How Many A's Do I Need to Raise My GPA?

Your grade point average or GPA is what universities and scholarship committees want to know when you're in high school applying to colleges for the first time. And once you're in college, it's what graduate school...

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### Advanced Lottery Math

Building on the previous math tutorial for calculating lotto odds, Basic Lottery Math, in this tutorial we will see how to calculate the probability of matching some of the numbers drawn and winning a lower-tier prize....

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### Basic Lottery Math

Most people who play the lottery know the odds against winning the jackpot are astronomically high; to invoke the popular adage that the lottery is a tax on stupidity may be a little harsh, as all lotteries are required...

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### Chord Length Formula for the Area of an Annulus

There are three formulas for calculating the area of an annulus using any two of the measurements for outer radius, inner radius, and thickness (width). However, there is another equation for the area of an annulus that...

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### Lucky for Life: Odds and How to Play

Lucky for Life originated as a Connecticut-only lottery in 2009, and in subsequent years became a multi-state lottery among six states in New England. In 2016 the game will expand to include at least 21 states spread...

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### Texas Two Step: Lottery Odds & Payout

Texas Two Step is a lotto draw game offered in the state of Texas. Tickets cost $1 for a single play and jackpots start of $200,000. If the jackpot is not won in a particular drawing, the unclaimed amount is added to...

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### Lotto Texas: Odds, Prizes, and How to Win

A quick-pick ticket for Lotto Texas with two number selections. Lotto Texas is a fun, multi-million dollar random number draw game offered by the Texas Lottery Commission. Lotto Texas offers a higher probability of...

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### Should You Play Powerball or Mega Millions? Probability & Expected Return

This article has been recently edited to reflect the most recent changes in both lotteries' structures. Powerball and Mega Millions are the two largest lotteries in the US in terms of the number of players, revenue...

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### Probability of Winning Florida Lotto with XTRA

Florida Lotto is one of seven draw games offered by Florida's lottery commission. After the multi-state lottery games Powerball and Mega Millions, Florida Lotto is the most popular jackpot game in the state. Drawings...

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### UK Lotto Odds | Lottery Probability

In the United Kingdom the biggest lottery after EuroMillions is the "Lotto," an easy to play game with jackpots in the several millions of pounds. A play slip for UK Lotto costs £2 and comes with...

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### Lottery Math: Fantasy 5 and Cash 5 Odds and Payouts

Odds and probability of winning lottery games known as Cash 5 and Fantasy 5, aka Pick 5 games. Explanation of odds and prizes for each state.

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### California SuperLotto Plus Odds & Payout

SuperLotto Plus is a popular lottery game in California with multi-million dollar jackpots and better odds of winning than Powerball or Mega Millions. A lottery ticket costs $1 for a single play and drawings are held...

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### New York Lotto vs. Sweet Million: Odds and Payouts

New York Lotto and Sweet Million are two random number drawing lottery games offered in the state of New York. While New York Lotto is still offered, the Sweet Million game was retired in the summer of 2014. The...

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### Keno Lottery Strategies, Odds, & Expected Winnings

Keno is a number drawing lottery game offered by many states. Keno games are fast since computerized drawings are held every 4 minutes or so and displayed on a monitor at the keno lottery vendor; players know within a...

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### New Jersey Pick-6 Lotto Odds & Payouts

The New Jersey Pick-6 lotto is a popular game in one of the most populous US states and costs just $1 to play. Jackpots start at a guaranteed $2 million, and frequently reach the tens of millions of dollars since New...

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### Ohio Classic Lotto: Odds of Winning and How to Play

Classic Lotto is an Ohio state lottery game, though residents from other states can buy tickets and claim prizes in Ohio. Each ticket costs $1 and the structure of the game is a typical pick-6-from-49 type of lotto. It...

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### Lottery Odds Formula

The chances of winning the lottery jackpot can be calculated with a straight-forward mathematical formula whose parameters depend on the structure of the lottery. Whether it's the PowerBall, Mega Millions, Lotto 6/49,...

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### Standard Deviation Made Easy -- Formula and Examples

Calculating standard deviation is easy. (source: FreePhotos) In a set of data, the standard deviation measures how spread out the numbers are with respect to the average (mean). A set with a high standard deviation is...

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### Volume & Surface Area of a Regular Pentagonal Pyramid

Equation to calculate the volume and surface area of a regular pentagonal pyramid using the length of one side of the base, and the length of the edge from the base to the apex.

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### Compare/Convert Norm-Referenced Test Scores with Mean & Standard Deviation

A norm-referenced test measures your skill in comparison to other test takers. Since scoring is relative, it matters how many questions you answer correctly and also how many questions others answer correctly. IQ tests...

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### Volume of a Parallelepiped from 4 Coordinate Points

Parallelepiped, see-through view. (source: author) A parallelepiped is a 3-dimensional geometric figure whose six faces are parallelograms. The faces come in three pairs; each member of the pair has the same size,...

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### Surface Area of a Conical Frustum (Truncated Cone)

A conical frustum (truncated cone) with height = H, bottom radius = B, and top radius = A. A truncated cone or conical frustum is a solid shape based on a cone, except that the pointed tip has been sliced off somewhere...

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### Hexadecimal (Base-16) Divisibility Rules

Base-16 (hexadecimal) is used in computing, for example, encoding colors in HTML. Divisibility rules are simple tests you can perform on a number to check if it is divisible by another number, without having to perform...

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### How to Draw Butterflies on a Graphing Calculator

Free drawings of butterflies generated by polar coordinate equations, how to draw butterflies on a graphing calculator. Symmetric and asymmetric butterfly curves.

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### Cow Tied to a Square Silo: Calculus/Geometry Problem with Solution

The classic cow and silo (or goat and silo) problem considers a cow tethered to a round silo with a rope that reaches half-way around the silo. The solution to that problem is explained in previous calculus tutorial. ...

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### Area and Arc Length of a Cardioid

A cardioid is a closed curve that resembles a heart with a rounded bottom instead of a pointed one. It also looks like the cross-section of a cherry. In Cartesian coordinates, the curve can be defined parametrically by...

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### Calculate the Median of a Continuous Probability Distribution

In a continuous probability distribution or probability density function p(x), the median is the value of x such that the area under curve to the left of x equals the area under the curve to the right of x. Since the...

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### Calculate Tree Height With Shadows

The shadow ratio method is one of the oldest and easiest methods to measure the height of a tree; all that is required is a measuring tape and a little math. This technique gives you the height of a tree using the...

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### Derivatives, Integrals, and Algebraic Identities for Square Roots

This clock should make sense to anyone who knows what a square root is. (FreeDigitalPhotos.net) Square roots (also called radicals) are one of the most fundamental concepts in mathematics and can be found in in almost...

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### Calculate Your Name Number in Numerology | Does It Mean Anything?

Can numerology predict your future? (Photo Source: Public Domain via Pixabay) Numerology, also called arithmancy, is the ancient practice of seeking out hidden meaning in words from numerical codes. Many cultures and...

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### Area and Perimeter of an L-Shape

An L-shape (also called an ell or ell-shape) is a common layout for rooms and outdoor areas. If you want to calculate how much trim you need to cover every side of an L-shape, you need to know the perimeter. To...

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### Turkey Roasting Time Calculator and Approximation Formula

The cooking time for a whole turkey is a function of the bird's weight and oven temperature. At a standard cooking temperature of 325°F (equivalently 163°C), the roasting time is more or less a linear function...

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### Inscribed Circles and Equilateral Triangles: 7 Hard Geometry Problems and Solutions

Geometry problems involving circles inscribed in equilateral triangles, and equilateral triangles inscribed in circles are often found in brain teasers and on standardized tests such as the SAT, GMAT, and GED. To...

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### Circles Inscribed in Squares: 7 Hard Geometry Problems & Solutions

Geometry problems involving circles inscribed in squares/rectangles, squares/rectangles inscribed in circles. Common on GMAT, SAT, and other standardized tests

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### Names of Parts of a Circle -- Radius, Diameter, Arc, Circumference, and More

A circle is defined as a closed curve with constant curvature, or alternately as a closed curve such that each point on the curve is equidistant to a locus in the interior of the curve. There are also names and...

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### Three Formulas for the Area of an Annulus & Examples

How to find the area of an annulus. In geometry, an annulus is ring-shaped figure bounded by two concentric circles. In other words, it is the space between two concentric circles. An annulus has three basic...

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### How to Tell If a Number Is a Pentagonal Number

How to tell if N is a pentagonal number. Formula for pentagonal numbers and inverse formula for determining if a given number is pentagonal. If yes, the formula gives the index.

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### How to Tell if a Number is a Fibonacci Number

Mathematical formula for testing whether or not a number is Fibonacci, and derivation of the formula. The test is an easier method than reconstructing Fibonacci sequence.

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### Irrational Numbers in Geometry

Irrational numbers, such as pi and most square roots, are numbers that cannot be expressed as fractions or repeating decimals. Their decimal expansions are infinite and never loop. When people first learn about...

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### Calculate the Thickness of Toilet Paper with Math

Correctly hung toilet paper. Source: Wikimedia Commons. If you're like me, you probably like to do math problems in your head while you sit on the throne. One of the easiest calculations you can do is figure the...

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### Probability of Rolling Sums on a Pair of D14 Dice (14-Sided Dice)

Dice with 14 sides are hard to find and mostly used by board game hobbyists. If you are mathematically curious, or just like collecting dice, you may own a few D14 dice even if you never play real games with them....

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### How to Integrate Sqrt(1 + Sqrt(x)) and Sqrt(x + Sqrt(x))

Nested radical functions such as y = sqrt(1 + sqrt(x)) and y = sqrt(x + sqrt(x)) have graphs that look much like those of simpler radical functions, such as y = x^(1/4) and y = sqrt(x), however, their...

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### How to Integrate Sqrt(x^2 - 1) and Sqrt(1 - x^2)

Typographically, the functions f(x) = sqrt(x^2 - 1) and g(x) = sqrt(1 - x^2) look very similar, but mathematically and graphically there are some key differences. The function f(x) = sqrt(x^2 - 1) is defined only for...

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### How to Integrate Sin(Sqrt(x)) and Cos(Sqrt(x))

Trigonometric functions of the form sin(x^p) and cos(x^p) are non-integrable unless the exponent p is the reciprocal of a positive integer. That is, if p = 1, 1/2, 1/3, 1/4, etc., then the antiderivative can be...

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### How to Solve Exponential Equations: Example Problems

Exponential equations arise in many areas of science, engineering, finance, and of course many branches of applied mathematics such as statistics. To solve exponential equations and word problems involving exponential...