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.
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!
