## Math Puzzles About Donkeys

Middle and high school level math and logic puzzles with a donkey theme.

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### How to Calculate the Distance Between a Point and a Sphere

The equation of a sphere in three-dimensional (x, y, z) coordinates is (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2 where the point (a, b, c) is the center of the sphere and r is the radius. For example, the surface...

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### Solar Panel Math: Calculate Energy Load and Array Size for a Photovoltaic System

Homeowners can reduce their long-run energy costs by installing photovoltaic systems to generate power for their homes. Solar panels can be set up completely off-grid to provide your home with energy independent of the...

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### Surveying Math: How to Calculate the Percent Grade of a Slope or Incline

An incline's grade (aka slope, gradient, or pitch) is the ratio of the change in elevation to the change in horizontal distance. Surveyors can compute topographic grade as a either a simple ratio or percent using...

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### Free M. C. Escher Tessellation Background Patterns: Tiling Lizard Background

Free M. C. Escher tessellations for websites. Maurits Cornelis Escher, more commonly known as M. C. Escher, was a graphic designer from the Netherlands who reinvigorated the art of optical illusions and tessellations....

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### How to Estimate the Cube Root of a Number with Calculus (Linear Approximation)

One useful application of calculus is to find the tangent line y = mx + b at given point on the graph of a function y = f(x) and use this linear equation to approximate values of the original function. To compute these...

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### Math Puzzle: Finding a Set of Four Consecutive Integers Given Their Sum

Given the sum of four consecutive integers, but not the four integers themselves, you can easily figure out what they are using elementary algebra. This type of problem is found on standardized math exams and in...

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### How to Count the Number of Spherical Candies in a Jar

Estimating the number of pieces of candy in a jar is a classic math problem, often used to test students' ability to use mathematics in real-life situations. If the candy is spherical and all the pieces are the same...

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### How to Calculate the Acreage of a Lot

An acre is an imperial and US customary unit of area, commonly used to describe the size of a piece of land or lot. It is defined as 1/640 of a square mile. its standard abbreviation is ac. Finding the acreage of a...

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### How to Measure the Thickness of a Sheet of Paper

The thickness of a single sheet of paper (known in the paper industry as its "caliper") is not something you can easily measure directly using a simple ruler. However, if you have a standard ream of paper (500 sheets)...

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### Math Puzzle: Finding Four Consecutive Integers Given Their Product

A staple of math exam questions and math puzzle collections is to find a set of consecutive integers whose sum or product is a given number. For example, finding four consecutive numbers whose sum is 3002, or four...

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### How to Compute Z-scores for a Normal Distribution

The standardized normal distribution has a mean of 0 and a standard deviation of 1. However, most populations in which a certain measurable characteristic is normally distributed don't have a mean of 0 and a standard...

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### Math Puzzle: How to Find 3 Numbers Given Their Average, Median, and Range

In statistics, three of the easiest set statistics to compute are the average (arithmetic mean), median, and range. The arithmetic mean or average of a set of numbers is the sum of all the numbers divided by the size...

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### How to Find the Volume or Capacity of an Octagonal Hot Tub

When you buy a new hot tub, the package will include the capacity of the tub in gallons or liters so you don't have to calculate anything. However, if you don't have the original package or specifications, you will have...

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### How to Calculate the Area of a Regular Hexagon

A regular hexagon is a polygon with 6 sides, all of which have the same length and meet at the same interior angle, 120 degrees. In order to find the area of a regular hexagon, all you need to know is the length of one...

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### Math Puzzle: How to Find 2 Numbers Given Their Sum and Product

A math brain teaser that can be solved with nothing more than high school algebra is finding the value of two numbers given their sum and their product. To solve this problem you need to set up a system of non-linear...

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### How to Compute the Probability of Throwing a Certain Sum on Three Dice

When rolling three dice, there are 16 different sums possible, every number from 3 to 18. Just as with throwing a pair of dice, certain sums are more likely to appear than others. For example, there is only one way to...

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### Calculate the Capacity of a Bookshelf or Bookcase

You can compute the capacity of a standard rectangular bookcase using simple arithmetic. All you need to know is the width of each shelf, the number of shelves, and the average width of a book in your collection. This...

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### How to Find the Distance to the Horizon at Sea Level

Use trigonometry to find the distance to the horizon (Image by Calculus-Geometry) The farthest point you can see when you stand at sea level with an unobstructed view is the horizon point. If you've every wondered...

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### How to Figure an NFL Quarterback's Passer Rating

The National Football League (as well as the Canadian Football League) assigns a passer rating to any player who has attempted at least one pass. The passer rating is a function of the number of attempts, completions,...

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### How to Compute the Probability of Rolling a Sum with Two Dice

Many board games and gambling games involve throwing a pair of dice whose sum determines your fate. Whether you're playing Monopoly, Parcheesi, or craps, you can get a better sense of what's going to happen if you know...

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### Basic Accounting Math: How to Calculate Straight Line Depreciation of Business Assets

When you record the value of your assets for accounting purposes, it is crucial that you account for depreciation, the loss of value over time. Any equipment that you purchase for your business--printers, computers,...

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### List of Trigonometric Identities, Derivatives, and Antiderivatives

Graphs of trig functions sin(x), cos(x), tan(x), etc. See more graphs, including arcsin, arccos, arctan... Trigonometric functions have many elegant properties and a number of surprising equivalent expressions, allowing...

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### How to Find the Volume of a Rectangular Prism (Rectangular Box) with Examples

A rectangular prism or rectangular box is a solid figure with six rectangular faces that meet at right angles. The top and bottom rectangles are the same size and shape and are parallel to each other, as are the...

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### How to Find the Surface Area of a Rectangular Prism (Rectangular Box) with Examples

A rectangular prism is a solid geometric figure with six rectangular faces that meet at right angles. The top and bottom rectangles are the same size and shape and are parallel to each other, as are the left/right and...

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### How to Find the Diagonal of a Rectangular Box (Rectangular Prism) with Examples

A rectangular prism is a solid geometric figure with six rectangular faces that meet at right angles. The top and bottom rectangles are the same size and shape and are parallel to each other, as are the left/right and...

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### Arc Length Integral for Polar Coordinates

In rectangular coordinates, each point on a curve is defined by its distance from the x- and y-axes. In polar coordinates, each point on a cruve is defined by the angle it makes with the origin and they positive x-axis,...

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### Area Integral for Polar Coordinates

Polar coordinates are an alternative parameterization of curves in two dimensions. Instead of representing a curve by a function y = f(x) where each point has the Cartesian coordinates (x, f(x)), polar equations are...

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### Integral for the Center of Mass of a Lamina (2-Dimensional Shape) Part 1: X-Coordinate

You can use integral calculus to find the center of mass or barycenter of a planar region bounded by defined curves. One integral gives you the x-coordinate of the center of mass, and another gives you the y-coordinate....

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### Integral for the Center of Mass of a Lamina (2-Dimensional Shape) Part 2: Y-Coordinate

Finding the center of mass (center of gravity or barycenter) of a planar region is a common application of integral calculus. The x- and y-coordinates of the barycenter of a 2-dimensional shape by evaluating two...

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### How to Find the Inverse of a Simple Rational Function (ax + b)/(x + c)

Given a function y = f(x), the inverse function is denoted by f -1(x) and satisfies the relations f(f -1(x)) = x f -1(f(x)) = x Also, if you graph a function and its inverse on the same graph with a square scale,...

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### How to Solve for the Principal in a Compound Interest Problem

In the compound interest formula A = P(1+r)^n, the variable P is the principal or initial amount that grows at a compounded rate. If you know the ending balance A, the periodic interest rate r, and the number of...

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### How to Solve for the Period in a Compound Interest Problem

In the compound interest equation A = P(1+r)^n, the amount earned is A, the initial amount is P, the interest rate as a decimal is r, and the number of compounding periods is n. A commonly encountered business and...

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### How to Solve for the Interest Rate in a Compound Interest Problem

The standard equation for compound interest gives the amount A that you can expect to receive at the end of n periods, given an initial quantity P and a periodic rate r. This equation is A = P(1 + r)^n, where r is given...

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### How to Find the Derivative of a Simple Rational Function (ax + b)/(x + c)

A rational function is a fraction with polynomials as the numerator and denominator. The simplest rational function is a ratio of two linear functions, (ax + b)/(x + c). Such functions are often encountered in applied...

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### Rational Function Antiderivative (ax + b)/(x + c)

Integrate a rational function. You will often encounter rational functions in physics, chemistry, engineering, and other fields that rely on math. Rational functions are ratios of polynomials where the degree of the...

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### How Fold a Cone With a Certain Height and Diameter

A cone is formed by folding a sector of a circle so that the two straight ends meet. If you want to make a cone that has a particular height and radius (half-diameter) then you need to carefully choose the shape and...

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### How to Calculate the Area of a Rhombus

A rhombus is a special type of quadrilateral in which all four sides are equal. This means that opposite sides are parallel and opposite angles are equal, so in fact, a rhombus is also a special type of parallelogram....

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### Proof of Heron's Formula (Area of a Triangle)

Heron's Formula is used to find the area of a triangle when the three side lengths A, B, and C are known. The formula is Area = sqrt[S(S-A)(S-B)(S-C)], where S = (A+B+C)/2.

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### Maximize Area of Rectangular Enclosure Against Two Perpendicular Walls

Fencing off a rectangular area adjacent to two perpendicular walls with a fixed length of material and maximizing the area of the enclosure is an example of how calculus can be used in practical applications. Suppose...

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### Maximize Area of Rectangular Enclosure Against a Wall

A simple real-world problem where calculus can help you find the ideal solution is building an enclosure with a fixed length of fence using a wall to form one of the sides. The goal is to maximize the total amount of...

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### How to Maximize the Volume (Capacity) of a Rectangular Box Cut From a Rectangle

A classic problem in calculus is the find the dimensions of a box with maximal volume that can be constructed from a rectangular piece of paper or stiff cardstock. The box is made by cutting four equally sized squares...

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### How to Represent a Number in Base-8 (Octal)

The octal numeral system has eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. The base-8 numbering system is sometimes used in computing instead of binary (base-2) or hexadecimial (base-16). The three are closely related since...

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### How to Convert a Number to Base-3

In the ternary (or trinary) number system, every number is represented as a string of 0's, 1's, and 2's. Base-3 numbering is not as frequently used as binary, octal, or hexadecimal, however it has some unique uses and...

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### Using the FOIL Method to Expand Products

The acronym "FOIL" is a mnemonic device to help you expand all the terms in a multiplication expression. F stands for first, O stands for outer, I stands for inner, and L stands for last. Of course you do not have to...

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### How to Find the Average Value of Function Over an Interval

The average value of a function is similar to the concept of the arithmetic mean of a finite set of numbers. When you want to know the average value of a function, you want to know the average height of each point on...

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### How to Convert Among Cubic Inches, Gallons & Liters

Volume conversion formulas for cubic inches, cubic feet, gallons, and liters. Examples of converting volumes from one unit to another.

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### How to Find the Base-16 (Hexadecimal) Representation of a Number Using the Remainder Method

The base-16 or hexadecimal number system is frequently used in computer science to encode values more efficiently than in binary (base-2) or decimal (base-10). For example, unicode characters are encoded in hexadecimal...

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### Convert to Base-2 (Binary) Using the Remainder Method

The binary numeral system, aka base-2, allows you to represent numbers using only two digits: 0 and 1. The simplicity of this number system makes it easier to add, subtract, and multiply numbers than with other bases,...

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### How to Solve Equations with Square Root and Linear Terms

Examples of radical equations with linear terms In algebra, radical equations are those that involve square roots, cube roots, fourth roots, etc. For equations that contain only square roots and linear terms, the...

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### How to Calculate the Area and Perimeter of a Circular Sector

how to figure the area and perimeter of a circular wedge or sector shape. equations and formulas for calculating area and perimeter

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### Mental Math: How to Quickly Multiply Numbers Using Square Numbers

There are many algebraic identities that give rise to mental math tricks. One of the most beautiful algebraic expressions that lends itself to multiplication and division is the difference of squares equation: x^2 -...

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### Volume of a Truncated Cone—Conical Frustum

Formula for the volume of a conical frustum (truncated cone). Volume = (pi/3)(h)(a^2+ab+b^2). Example of how to compute the volume of a paper cup shaped like a conical frustum.

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### Practical Math Guide: Calculating Your Car Lease Payments

Formula for calculating monthly car lease payments as a function of the price, residual value, length of the lease, and money factor.

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### How to Tell if (x-a) Is a Factor of a Polynomial

How to determine if a binomial factors into a polynomial, applying the fundamental theorem of algebra...

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### Calculate the Volume of a Sphere from Its Circumference

You know spherical volume in terms of the radius, but what is the volume in terms of the circumference? v = c^3/(6pi^2). See derivation and example.

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### Cylindrical Shells Method for the Volume of a Solid of Revolution

How to find the volume of an object with circular cross-sections (solid of revolution) using the cylindrical shell method in calculus.

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### Disk Method for the Volume of a Solid of Revolution

How to calculate the volume of an object with circular cross-sections (solid of revolution) using the calculus disk method.

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### Solid of Revolution: Exponential Function Around the Horizontal Axis (X-Axis)

How to find the volume of a solid of revolution (around the x axis) using an exponential function as an example.

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### Solid of Revolution: Exponential Function Around the Vertical Axis (Y-Axis)

How to find the volume of a solid of revolution (around the y axis) using an exponential function as an example.