related rates fence problem

Question

A fence is to be built to enclose a rectangular area of 220 square feet. The fence along three sides is to be made of material that costs 5 dollars per foot, and the material for the fourth side costs 14 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

Answer 1

3(5)x +14y = total Cost = C

xy = 220

x = 220/y

15x +14y = C

C = 15(220)/y +12y

C'(y) = -15(220)/y^2 +14 = 0

multiply by y^2

-15(220) +14y^2 = 0

y^2 = 3300/14 = 11(300)/(14)

y^2=11(150/7

y^2=1650/7 = 235 5/7

y = sqr(1650/7) = about 15.353 feet for the $12 side

x = 220/15.353 = about 14.329 feet for each of the three $5 sides

an alternative method is

graph the Area function. It's a rectangular hyperbola, 220 = xy

with x= 220/y

at the point where x=y, the slope = -1 and x=y=sqr220 =

the Cost function C= 15x +14y could be rewritten

as 14y =-15x +C

or y = -15x/14 + C/14 which has slope = -15/14

find the point on the hyperbola which has slope =-15/14

that (x,y) will tell you how much of each side to use to minimize cost

-15/14 is almost -1, so the lengths of the sides will almost be equal