Please solve this, i'm desperate

Question

Jake is building a fence around his property. He wants the perimeter to be no more than 100 feet. He also wants the length to be at least 20 feet longer than the width. If he builds his fence according to these limits, which would be the maximum possible width of the fence?

Answer 1

Hi Paige!

We can start by writing a few equations to get the ball rolling.

First, let's let L=the length and W=the width.

We know the entire perimeter must be less than or equal to 100 ft, so

2L + 2W ≤ 100

or

W ≤ 50 - L

Then, we know that the length needs to be at least 20 feet longer than the width, so

L ≥ W + 20

or

W ≤ L - 20

Putting it all together, our problem can be written as follows

Maximize W

Subject to W ≤ 50 - L

W ≤ L - 20

Since both of these are LESS THAN or equal to inequalities, the maximum will be at equality.

So, that means

W = 50 - L

and

W = L - 20

Which is two equations with two unknowns! A simple system of equations.

If we solve the first equation for L, we have

L = 50 - W

then, we substitute into the second equation

W = 50 - W - 20

or

2W = 30

or

W = 15

SO, the maximum width is 15 feet.

Answer 2

Assign variables: L = length, and W = width

Write equations:

I) 2L + 2W ≤ 100

II) L - W ≥ 20

First consider the special case where each relationship is simply =

2L + 2W = 100

L - W = 20

Solve this pair to get L = 35 and W = 15

Next, try

W = 20. Using the 2nd equation, this means that L must be at least 40.

In the first equation 2L + 2W = 120, which is too large. Therefor, I think 15 is the maximum for W