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Solving Word Problems Involving Quadratic Functions

Updated on April 27, 2011

Solving Word Problems Involving Quadratic Function


Some of the most important functions in applications are the quadratic functions. The quadratic functions are one of the simplest type of polynomial function, aside from the linear functions. Applications include satellite dishes, projectile paths, economics, geometry, ecology, weather,biology and many others.


A function f is a quadratic function if f(x) = ax2 + bx + c where a, b, and c are real numbers and not equal to 0.. The graph of a quadratic function is a parabola. The most basic aid in graphing a parabola is knowing whether a > 0 (the graph opens upward) or a < 0 (the graph opens downward). The two simplest quadratic functions are f(x) = x2 and g(x) = - x2 .


In this hub, I feature two interesting problems with their complete solution.


Problem Number One :


Jack Pott dives off the high diving board. His distance from the surface of the water

varies quadratically with the number of seconds that have passed since he left the

board.


(a) His distance at time 1, 2 and 3 seconds since he left the board are 24, 18, 2

meters above the water respectively. Write a particular quadratic equation

expressing the distance in terms of time.

(b) How high is the diving board ?

(c ) When does he hit the water ?


Solution :

(a) In finding the quadratic equation three points can be derived from the given :

(1, 24 ) ; (2, 18) ; (3, 2);

by letting the time in seconds as x-coordinates and the distance above the

water as y-coordinates.

The quadratic equation has a standard form ax2 + bx + c = 0.

Using the given points above, we must solve for a, b, and c.

From the point (1, 24) we can derive an equation by substituting 1 to x’s and

24 to y, The equation formed is : a + b + c = 24 let this be equation #1.

From point (2, 18) we can derive this equation : 4a + 2b + c = 18 let this be

equation #2.

From point ( 3, 2 ) we can derive this equation : 9a + 3b + c = 2 let this be

Equation #3.

We will now solve for a, b, and c using elimination method. by subtraction.

Using equation 1 and equation 2 to eliminate c ;

a + b + c = 24

-4a - 2b - c = -18

__________________________________

-3a - b = 6 let this be equation 4

Using equation 2 and equation 3 to eliminate c

4a + 2b + c = 18

-9a - 3b + c = - 2

-5a - b = 16 let this be equation 5

Solving for a and b using equation 4 and equation 5

-3a –b = 6

+5a +b = -16

-------------------------------------

2a = -10

a = -5

substituting a = -5 in equation 4

-3(-5) – b = 6

15 – b = 6

b = 15 – 6

b = 9

substituting a = -5 b = 9 in equation 1 to solve for c

-5 + 9 + c = 24

c = 24 – 4

c = 20

Therefore the quadratic equation we are looking for is -5x2 + 9x + 20 = 0.


(b) How high is the diving board ?

In order to solve for the height of the diving board we must solve for the vertex of the parabola y = -5x2 + 9x + 20.. This parabola opens downward since a is negative.

Actually, the vertex is the maximum point of the parabola. The height of the diving board is the y-coordinate of the vertex. Solve first for h or x-coordinate of the vertex . The formula to solve for h is -b/2a. Therefore h = -9/(2) (-5) = 9/10.

Substitute 9/10 to every x in the given quadratic function y = -5x2 + 9x + 20.

y = -5 (9/10)2 + 9 (9/10) + 20

y = -81/20 + 81/10 + 20

y = 481/20 = 24.05 meters.


( c ) When does he hit the water ?

y = 0, when the diver hits the water so in order to solve for this we must let

-5x2 + 9x + 20 = 0 and solve for x using quadratic formula.

The quadratic formula is given by :

X = ( -b + - SQRT(b^2-4ac) )/2a

X = (-9 + - SQRT(81 – (4)(-5)(20)) ) /2(-5)

X = (-9 +- SQRT(481) )/-10

X = ( -9 – 21.93 ) /-10

X = -30.3/-10

X = 3.03 seconds


Problem Number Two :

A satellite dish in storage has parabolic cross sections and is resting on its

vertex.. A point on the rim is 4 feet high and is 6 feet horizontally from the

vertex. How high is a point which is 3 feet horizontally from the vertex?


Solution :

Plotting this parabola on a rectangular coordinate plane with vertex at (0, 0 ), we can derive other two points based from the given . Two points on the rim are

(-6, 4) and (6, 4).

Now let us derive the equation of the parabola using points (0, 0) ; (-6, 4), (6, 4).

C = 0

Using point (-6, 4 ) and (6, 4)

4 = a (-6)^2 + (-6)b + c

4 = 36 a -6b + c

4 = 36 a + 6b + c

========================

8 = 72 a

a = 1/9

Solving for b ;

4 = 36 (1/9) + 6b

4 = 4 + 6b

B = 0

The equation of the parabola is y = 1/9 x^2

Y = 1/9 (3)^2

Y = 1/9(9) = 1

The point which is 3 feet horizontally from the vertex is 1 foot high.

Comments

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    • profile image

      rcapshaw 

      5 years ago

      I don't think your height of the diving board is correct. You found the max height the diver gets above the water. Wouldn't the diving board height be when t = 0? I think the diving board is 20 meters and the diver reaches the height of 24.05 meters/.9 seconds after leaving the board.

    • profile image

      aspicephype 

      5 years ago

      When i employed to find on top of existence although recently We've established a new opposition.

    • profile image

      ferlyn 

      5 years ago

      for me nice subject that:P)

    • profile image

      Unknown person 

      5 years ago

      Given Problem:

      A cellular phone store can sell 100 phonekits per month for P5,000 each. Reseach indicates that the store can sell 50 more kits per month for each P100 decrease in price. What price per kits per month give greatest monthly sales? What is the sales amount?

      Can you explain the answer and what is the mean of "50 more and P100 decrease".

    • profile image

      Christopher 

      6 years ago

      I love math

    • math-worksheets profile image

      math-worksheets 

      6 years ago from California

      Cristina, great great and once again great! That is what I'm looking for! :) Thanks!

    • profile image

      Guillermo Bautista 

      6 years ago

      I'm glad because only few Filipinos are blogging about math. I also have mine at http://mathandmultimedia.com/.

      Keep up the good work Cristina.

    • profile image

      zomarie 

      6 years ago

      very good answers

    • profile image

      lhei 

      7 years ago

      this hub is si nice.:)

      it helps me a lot .

      thanks for creating this beautiful hub :)

      --

      more blessings 2 come . god bless.

    • profile image

      sushmitha 

      7 years ago

      thanks!

    • cristina327 profile imageAUTHOR

      Maria Cristina Aquino Santander 

      7 years ago from Manila

      Hi Dave Matthews it is good to see you visiting this math hub. Thank you for dropping by. Regards.

    • Dave Mathews profile image

      Dave Mathews 

      7 years ago from NORTH YORK,ONTARIO,CANADA

      Ate Cristina, thanks for this Hub. I'm afraid you lost me though for when it comes to math 1 + 1 = 11 I get lost. I do appreciate learning though.

      KUYA Dave.

    • cristina327 profile imageAUTHOR

      Maria Cristina Aquino Santander 

      7 years ago from Manila

      Hi RevLady it is great to see you visiting this mathematical hub. Thank you for gracing this hub. Your visit and comment is much appreciated. I am glad ypu find this interesting. Blessings to you.

    • RevLady profile image

      RevLady 

      7 years ago from Lantana, Florida

      Thank you Cristina for carrying me out of my comfort zone into a realm of which I know very little. Quite interesting if not challenging.

      God bless you,

      Forever His,

    • cristina327 profile imageAUTHOR

      Maria Cristina Aquino Santander 

      7 years ago from Manila

      Hi torconstantino it is good to hear from you. Thank you for taking time to read this hub. Your visit and comment is much appreciated. Blessings.

    • torconstantino profile image

      torconstantino 

      7 years ago from Maryland

      Wow! I'm glad there are brains like yours that are big enough to wrap around quadratic word problems - thanks for sharing here!

    • cristina327 profile imageAUTHOR

      Maria Cristina Aquino Santander 

      7 years ago from Manila

      Hi lifegate glad to see you visit this mathematical hub. Hope you enjoyed what you have read. Thank you for gracing this hub. It is much appreciated. Blessings.

    • lifegate profile image

      William Kovacic 

      7 years ago from Pleasant Gap, PA

      Cristina, I have to admit math was never one of my better subjects. Glad you have it all together.

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