Question

The back of jake's property is a creek. Jake would like to enclose a rectangular area, using the creek as one side and fencing for the other three sides, to create a pasture. If there is 460 feet of fencing available, what is the maximum possible area of the pasture.

Answer 1

Answer:

The maximum possible area of the pasture  is 26,450 feet square.

Step-by-step explanation:

Let us assume the width of the rectangular area  = m  ft

and the length of the area = k ft

Also assume:  ( one side with k units IS NOT FENCED)

So, the total  perimeter of the fencing is:

2 m + k = 460

or, k = 460 - 2m  ........ (1)

Now, AREA OF THE RECTANGULAR PORTION = Length x  Width  = k x m

Put  k =  460 - 2m

We get:   A = m (460 - 2m)

or, A =  460 m  - 2m²

We need to find the maximum for the parabolic function A = 460 m  - 2m²

The function has a maximum value as the quotient in front of x^2 is negative: -2 < 0

A (max)  = $c-\frac{b^2}{4a}$   where a = -2, b = 460, c = 0

= $-\frac{(460)^2}{4(-2)} = 26,450$

A max = 26,450 sq ft.

The maximum possible area of the pasture  is 26,450 feet square.