Question

A farmer wishes to enclose a rectangular pen with area 100 square feet next to a road. The fence along the road is to be reinforced and costs $34 per foot. Fencing that coast $16 per foot can be used for the other three sides. What dimensions for the pen will minimize the cost to the farmer. What is the minimum cost?

Answer 1

1X100

Answer 2

let the length be x, let the width be y

xy = 100
y = 100/x

cost = 34x + 16x + 16(2y) = 34x + 48y
= 34x + 48(100/x)

d(cost)/dx = 34 - 4800/x^2 = 0 for a minimum cost
34 = 4800/x^2
x^2 = 4800/34
x = appr. 11.88 ft

pen is 8.42 by 11.88 ft, with the 11.8 ft along the road


minimum cost = 34x + 4800/x = 807.96

check:
take x = 12 , cost = 808
take x = 11 , cost = 810.36
take x = 1 (Steve's answer) , cost = 4834
take x=100 , cost = 3448

The answer of 11.88 by 8.42 is correct for a min cost of $807.96

Answer 3

I was joking with the 1x100. Also, I notice that your solution is incorrect, because 16(2y) is not 48y.

The correct solution has appeared elsewhere as 8x12.5