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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his bam. 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 $20 per linear foot to install and the farmer is not willing to spend more than $5000, find the dimensions for the plot that would enclose the most area.

To determine

To find: The dimensions of the rectangular plot that would enclose the most area.

Explanation

Let the length of the rectangular plot be x ft and the width of the rectangular plot be y ft.

In Figure 1, the barn is on the north side of the rectangular plot.

On the north side, no fencing is needed.

On the west side, fencing is needed for the length x ft.

The fencing costs $20 per linear foot to install.

Therefore, the fencing cost for the west side is $20x.

Since the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence.

Hence, the farmer has to spend $20x2=$10x.

On the south and east side together, the farmer has to fence (x+y)ft.

So the cost is $20(x+y).

Therefore, the total cost for fencing is $(10x+20x+20y).

Since, the farmer is not willing to spend more than $5000, the equation is,

(10x+20x+20y)=500030x+20y=500020y=500030xy=25032x

The area of the rectangular plot is,

A=lengthwidth=xy

Substitute y=25032x in the area A=xy,

A=x(25032x)=250x32x2

Differentiate A with respect to x,

dAdx=2503x

For critical points, dAdx=0

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