Question

A dog kennel with four pens is to be constructed. The pens will be surrounded by rectangular fence that costs $23 per meter. The rectangle is partioned into four pens of equal size with three partitions made of fence that costs $12 per meter. Each pen measures x meters wide by y meters long, as in the figure.What is the total cost of the fence?Suppose each pen must have 14 square meters of area. What should x and y be to minimize the cost of the fence?

Answer 1

outside perimeter is 2(4x+y) = 8x+2y
inside sections use 3y

cost is thus

c = 23(8x+2y) + 12(3y)
= 184x + 82y

Now, if the pens each occupy 14m^2, then y = 14/x, so

c = 184x + 82(14/x)
= 184x + 1148/x

minimum cost where dc/dx=0, or

184 - 1148/x^2 = 0
x = √(287/46) = 2.497

Now just figure y