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 - bobpursley, Wednesday, October 12, 2011 at 8:14pm
Area= R*W where road fence is R, W is the length perpendicular to the Road.
Cost= 34R+16(R+2W)= 34Area/W+ 16(Area/W+2W)
dCost/dw= -34*100/W^2+ 16(-100/W^2+2)=0
W= sqrt (5000/32)
check my math.
calc - Steve, Wednesday, October 12, 2011 at 8:16pm
let there be length a and width b, with side a along the road.
b = 100/a
cost is a*34 + a*16 + 2*100/a * 16
c = 50a + 200/a
c' = 50 - 200/a^2
c' = 0 when a = 2
so, the minimum cost is 100 + 100 = 200
a 2' wide pen? Is he housing gerbils?
calc - Steve, Thursday, October 13, 2011 at 12:07am
My bad - bobpursley is correct. The road length is 8, width is 12.5
using my notation,
c = 50a + 3200/a