Minimizing Construction Costs

Question

The management of the UNICO department store has decided to enclose an 836 ft² area outside the building for displaying potted plants and flowers. One side will be formed by the external wall of the store, two sides will be constructed of pine boards, and the fourth side will be made of galvanized steel fencing. If the pine board fencing costs $7/running foot and the steel fencing costs $4/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (Round the answers to one decimal place.)

Answer 1

Area= A = 836 ft^2

A =Length times Width = LW

L = A/W = 836/W

Perimeter = 2W + L

Cost = 7(2W) + 4L

$7 per foot for pine,

$4 per foot for steel

Cost = C = 14W + 4(836/W)

C = 14W + 3344/W

take the derivative and set equal to zero, to find cost minimizing dimensions

C' = 14 -(3344)/W^2 = 0

multiply by W^2

14W^2 = 3344

W^2 = 3344/14 = 1672/7

W =sqr(1672/7) = about 15.46 feet = 15.5 ft width

L = 836/15.46 = about 54.09= 54.1 ft long

dimensions are 15.5 by 54.1 feet

Minimum Cost = 7(15.5) + 4(54.1)

= 108.5 + 216.4

=about $324.90