Enlarging the Penguin Pond

Date: 06/03/99 at 14:14:11
From: Crystal
Subject: Quadratic equation

Hi! I'm not at word problems; they seem easy but they are hard for me 
to solve.

Here is the problem: The rectangular penguin pond at the Bay Park zoo 
is 12 meters long by 8 meters wide. The zoo wants to double the area 
of the pond by increasing the length and width by the same amount. By 
how much should the length and width be increased?

Date: 06/07/99 at 12:39:20
From: Doctor Arthur
Subject: Re: Quadratic equation

Crystal, 

Don't feel bad about word problems; most people don't do well with 
them. 

I suggest that the best way to do this problem would be with a chart 
so that you can see why they give you certain information. Here:

     ---------------------------   Since the zoo wants to increase
     |Length|Width|    Area    |   both the length and the width by
     ---------------------------   same amount, we'll use the variable
 Now |  12  |  8  |  96        |   x to represent the amount being 
Later| 12+x | 8+x |(12+x)*(8+x)|   added to both.
     ---------------------------   (Note: Since x is a length, x>0.)

Since we know that the area later will be twice the current area, we 
can write the following equation:

     (12+x)*(8+x) = 2(96)

By multiplying out and getting everything on one side, we get

      x^2 + 20x - 96 = 0   which factors into 

         (X+24)(x-4) = 0 

Using the zero product property,

     x = -24 and x = 4

Since x can't be negative, therefore x = 4. So if the zoo adds 4 
meters to the length and the width, the area of the new pond would be 
double the area of the original.

I hope this will help you out. Come back again if you need more.

- Doctor Arthur, The Math Forum
  http://mathforum.org/dr.math/   

Date: 06/03/99 at 17:00:56
From: Doctor Rob
Subject: Re: Quadratic equation

Thanks for writing to Ask Dr. Math!

Here are some tips on solving "Word Problems."
--------------------------------------------------------------
Solving an Applied Problem

First convert the problem into mathematics. This step is (usually) the 
most challenging part of an applied problem. If possible, start by 
drawing a picture. Label it with all the quantities mentioned in the 
problem. If a quantity in the problem is not a fixed number, name it 
by a variable. Identify the goal of the problem. Then complete the 
conversion of the problem into math, i.e., find equations that 
describe relation among the variables, and describe the goal of the 
problem mathematically.

Solve the math problem you have generated, using whatever skills and 
techniques you need.

As a final step, you should convert the answer of your math problem 
back into words, so that you have now solved the original applied 
problem. 
---------------------------------------------------------------

Let's apply these steps to the zoo problem. We start by drawing a 
picture:

   G                               F
    o-----------------------------o
    |                             |
    |                        C    |
  D o-----------------------o     |
    |                       |     |
    |                       |     |
    |                       |     |
    |                       |     |
    o-----------------------o-----o
   A                        B      E

Here rectangle ABCD is the original pond. The problem tell us that AB 
= CD = 12 meters, and BC = AD = 8 meters, so we label those parts with 
those dimensions:

   G                               F
    o-----------------------------o
    |                             |
    |                        C    |
  D o-----------------------o     |
    |          12           |     |
    |8                     8|     |
    |                       |     |
    |          12           |     |
    o-----------------------o-----o
   A                        B      E

Then we are told that the zoo wants to double the area. We know the 
formula for the area of a rectangle is A = L*W, where L is the length 
and W is the width, so we can find the area of the original pond: A = 
8*12 = 96 square meters.

The new pond will be longer by an amount we don't know but want to 
find. Let's call that x meters, following the idea of naming 
quantities we don't know with variables. That means that the new pond 
length AE = FG = 12+x meters. We are also told that the width of the 
pond is increased by the same amount, that is, x meters, so the new 
pond width is 8+x meters. We label BE and DG with x, and EF with 8+x 
and FG with 12+x:

   G              12+x             F
    o-----------------------------o
   x|                             |
    |                        C    |
  D o-----------------------o     |
    |          12           |     |8+x
    |8                     8|     |
    |                       |     |
    |          12           |  x  |
    o-----------------------o-----o
   A                        B      E

Now there is one last part of the problem statement: the part which 
will give us an equation to solve. That is, that the area of the new 
pond (which we know is its length times its width) is double that of 
the old pond (which we figured was 96). Here is the equation:

     (12+x)*(8+x) = 2*96.

Now we have finished translating the problem into mathematics and we 
use the appropriate math technique to solve this equation. In this 
case, when you multiply everything out (using FOIL, for example) and 
collect like terms, you will have a quadratic equation. Choose your 
favorite way to solve quadratic equations and find the values of x 
which make this equation true. There are two, because quadratic 
equations always have two solutions.

Now we have to translate the answers back into a solution of the word 
problem. One of the values of x will be negative, and that will make 
no sense in this problem, so that value of x should be discarded. The 
remaining solution will be given in the form, "The zoo should make the 
pond __ meters longer and wider." Fill in the blank with that value of 
x.

Understand?

The same techniques can be used on essentially all "Word Problems."

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/