Posted by Andres on Thursday, October 13, 2011 at 12:04am.
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?
- Calc - Steve, Thursday, October 13, 2011 at 12:36am
- Calc - Reiny, Thursday, October 13, 2011 at 8:19am
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
- Calc - Steve, Thursday, October 13, 2011 at 10:10am
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
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