Question

The management of a local Target store has decided to enclose a 500 square foot area outside the building for the garden display. One side will be formed by an external wall of the store, two sides will be constructed of pine boards costing $6 per foot and the side opposite the store will be constructed of fencing that costs $3 per foot. What dimensions of the enclosure will minimize the cost? Let x be the length of the side with fencing and y be the length of the sides with pine boards.

Answer 1

Explanation:

the area of a rectangle is

length × width

so, in our case

x × y = 500 ft²

x = 500/y

3x + 2×6×y to be minimum.

using the upper identity in the minimum expression gives us

3(500/y) + 12y

1500/y + 12y

we find the extreme points by finding the zeros of the first derivative :

(1500/y + 12y)' = 0

-1500(y^-2) + 12 = 0

-1500/y² + 12 = 0

12 = 1500/y²

12y² = 1500

y² = 125 = 25×5

y = ±5×sqrt(5)

a negative length does not make sense, so

y = 5×sqrt(5) ft

is our solution.

x = 500/y = 500 / (5×sqrt(5)) = 100/sqrt(5) =

= 20×5/sqrt(5) = 20×sqrt(5)×sqrt(5)/sqrt(5) =

= 20×sqrt(5) ft