Question

A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $8 per linear foot to install and the farmer is not willing to spend more than $4000, find the dimensions for the plot that would enclose the most area. (width, length)

Answer 1

Answer:

x = 125 ft and y = 250/3 ft

Step-by-step explanation:

Let assume that,

x be the length of the northern part of the fence (parallel to the north wall)

y be the length of the western and eastern pieces of the fence

As well as farmer has $4000 to spend hence we can write,

8x + 8y + 4y = 4000

8x + 12y = 4000

Hence we can say that,

from the above equation we can write

$y=\frac{4000}{12} -\frac{2x}{3}$

We know that area

A=xy.

$A(x) =\frac{500x}{3}-\frac{2x^2}{3}$

we can write

$A'(x) = \frac{500}{3} -\frac{4x}{3}$

equating it 0 we get

$A'(x) = \frac{500}{3} -\frac{4x}{3}=0$

$x=125$

Also,

$A"(x) = -\frac{4}{3}$

which is less than zero.

we can see that A''(x) is always less than 0 hence using second derivative test we can say that x = 125 is a maximum point.

now, solving for y we get  $y= \frac{250}{3}$

Hence we can say that dimensions for the plot that would enclose the most area is,

x = 125 ft and y = 250/3 ft