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### Enlarging the Penguin Pond

```
Date: 06/03/99 at 14:14:11
From: Crystal

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

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

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
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/
```
Associated Topics:
High School Basic Algebra
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons

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