posted by Andres on .
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?
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
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
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