Hagrid wants to build some pens for his creatures.

Question

The next day

Hagrid wants to build some pens for his creatures. He wants to build one rectangular

enclosure that is divided into three equal parts. If he

has 1000 ft. of fencing to use,

what is the largest

area he can enclose?

Answer 1

Convince yourself that a rectangular area with 3 fenced spaces will have the following constraint equation:

Constraint equation: 4x + 2y = 1000

Optimization equation: xy = A

Goal: maximize A (think Calculus, think derivatives)

In order for us to take a derivative, we need to get the optimization equation in terms of a single variable. So, let's take our constraint equation and solve for y.

4x + 2y = 1000 ==> 2y = 1000 - 4x ==> y = 500 - 2x

Note: ==> stands for "implies".

Now, let's go to our optimization equation and substitute y with 500 - 2x. When we do, we get:

x(500 - 2x) = A ==> 500x - 2x^2 = A

Let's now take a derivative:

A' = 500 - 4x

Now, an important step: we need to set the equation to 0.

0 = 500 - 4x ==> 500 = 4x ==> x = 125

Now that we know x, let's calculate y by going to our constraint equation:

4 * 125 + 2y = 1000 ==> 500 + 2y = 1000 ==> 2y = 500 ==> y = 250

And now that we know x and y, we can calculate the maximum area

A = xy = 125 * 250 = 31250 sq.ft.

Answer 2

Draw and label a diagram!

w represents the width and the two interior walls

(1000 - 4w) / 2 represents the length, or 500 - 2w

A = w(500 - 2w)

A = 500w - 2w²

Maximum at w = -500 / 2(-2), or 125

Length is 250

Area is 31,250 ft²