   Chapter 3.7, Problem 13E

Chapter
Section
Textbook Problem

# A farmer wants to fence in an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?

To determine

To minimize:

The cost of the fence

Explanation

1) Concept:

Length of fencing is nothing but the sum of all lengths and widths

First derivative test for absolute extreme values- suppose that c is a critical number of a continuous function f defined on an interval.

a) If f'x>0 for all x<c and f'x<0 for all x>c, then fc is the absolute maximum value of f

b) If f'x<0 for all x<c and f'x>0 for all x>c, then fc is the absolute minimum value of f

2) Given:

Total area of the rectangular field 1.5 million

3) Formula:

Area of rectangle =l·h

4) Calculation:

Let m and n be the lengths (in feet) of the sides of rectangle, also the total area of the rectangle is 1.5  million that is 1500000 ft2

Area =mn=1500000

The total fence length is made up of 2  sides of length m and 3 sides of length n

F=2m+3n This is the equation we want to minimize

We have mn=1500000

Divide both sides by m

n=1500000m

Now put n=1500000m is in F=2m+3n

=2m+3(1500000m)

F=2m+4500000m-1

Now, differentiate F using power rule

F'=21+(4500000·-1·m-<

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