A fence must be built to enclose a rectangular area of 20 comma 000 ftsquared. fencing material costs $ 2 per foot for the two sides facing north and south and $4 per foot for the other two sides. find the cost of the least expensive fence.

The first thing we should do is write the functions.

For the area we have:

A = w * h = 20000 feet ^ 2

The price is:

Cost = (4 * 2 * h + 2 * 2 * w)

Substitute for h ::

C (w) = (4 * 2 * (20,000 / w) + 2 * 2w)

Rewrite we have:

C (w) = 160,000 / w + 4w

Take the derivative:

C ' (w) = 160000 (-1 / w ^ 2) + 4

We equal zero and clear w:

0 = 160000 (-1 / w ^ 2) + 4

160000 / w ^ 2 = 4

w ^ 2 = 40000

w = 200 (approximately)

width = 200 ft; so cost is 2 * ($ 4) 200 = $ 1600

height = 20000/200 = 100 ft; so cost is 2 ($ 2) 100 = $ 400

Total cost = $ 2000

Answer:

The cost of the least expensive fence is:

Total cost = $ 2000