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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

If the farmer in Exercise 18 wants to enclose 8000 square feet of land, what dimensions will minimize the cost of the fence?

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 bam, 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 that will minimize the cost of the fence

Explanation

1) Concept:

i. A critical number of a function f   is a number c in the domain of f  such that either  f'c=0 or f'c does not exist.

ii. The First Derivative Test-Suppose that c is a critical number of a continuous function f

a) If f' changes from positive to negative at c, then f has a local maximum at c

b) If f' changes from negative to positive at c, then f has a local minimum at c

c) If f' is positive to the left and right of c, or negative to the left and right of c, then f has no local maximum or minimum at c

2) Formula:

Area of rectangle=length· height

3) Given:

Fencing cost is $20 per linear foot, and the farmer is willing to spend maximum $5000 for that, also, the farmer wants to enclose 8000 square feet of land

4) Calculation:

The fencing cost $20 per linear foot to install and the cost of the fencing on the west side will be split with the neighbor, so the farmer’s cost C will be

C=1220x+20y+20x=20y+30x

The area A to be enclosed is 8000 ft2,

So A=xy=8000

y=8000x

Now

C=20y+30x

Substitute y=8000x

C=208000x+30x=160,000x+30x

Differentiate C with respect to x,

C'=-

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