30000 feet and wants to make a feild and divide it into 2 equal pastures

Question

a rancher has 30000 linear feet of fencing and wants to enclose a field then divide it into three equal peices with 2 internal fences parallel to one of the rectangular sides. what is the maximum area of each individual area

Answer 1

Let the sides of the fence be x and y.

We have: 30000 = 2y + 4x

Rearranging, we will get:
15000 = y + 2x
y = 15000 - 2x

Let the area of entire fenced pasture be A.
Thus, A = xy.

Substituting y = 15000 - 2x in, we will get:

A = x(15000 - 2x)
A = 15000x - 2x^2

Diferentiating this, we will get:

A' = 15000 - 4x

Setting the derivative to zero, we will have:

A' = 0
15000 - 4x = 0
4x = 15000
x = 15000 / 4
x = 3750

Thus, the maximum area is achieved when x = 3750.
When x = 3750, y = 15000 - 2(3750) = 7500

This means the maximum area is 7500 * 7500 = 56250000 feet^2

Hence, the maximum area of each individual area is:
56250000 / 3 = 18750000 feet^2