G a fence must be built to enclose a rectangular area of 5000 ft^2. 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.

Let Width = W, then Length = L = 5000/W

Apply the $4 fence on the shorter side W, then total cost

C (W) = $4 * (2W) + $2 * (2L)

= (8W^2+20000) / W

To find the minimum cost, differentiate C (W) with respect to W and equate to zero and solve for W:

C' (W) = (8W^2-20000) / W^2=0 = > 8W^2-20000=0 = >

W=sqrt (20000/8) = 50'

Length L=5000/50=100'

We now need to check that W indeed gives a minimum cost C (W).

This can be done by checking the sign of C" (W), the second derivative.

The second derivative C" (W) = 40000/W^3 >0, which means that W=50 gives a minimum value for C (W).

The total (minimum) cost is therefore

C (50) = (8*50^2+20000) / 50=$800

check average cost = $800 / (2 (50+100)) = $800/300=$2.67 / foot,

which is between $2 and $4, and that sounds reasonable!