A rectangular building with a square front is to be constructed of materials that costs 13 dollars per ft2 for the flat roof, 12 dollars per ft2 for the sides and the back, and 17 dollars per ft2 for the glass front. We will ignore the bottom of the building. If the volume of the building is 5,600 ft3, what dimensions will minimize the cost of materials?

The front side is 11.483 ft while the width of the building is 42.473 ft

Step-by-step explanation:

Volume of the building = 5600 ft³

Area × Width

Side² × Width

Roof dimension = Side × Width

Dimension of side = Side × Width

Dimension of front = Side × Side

Therefore we have, the cost given as follows;

Let the side = S and the width = W

Cost therefore = 13(W·S) +12(2W·S + S²) + 17(S²)

Cost therefore =29·S²+37·W·S

Therefore, we are to minimize 29·S²+37·W·S subject to W·S² = 5600

Which gives

58·S + 37·W = λₓ

37·S =  λ

58·S + 37·W = 37·S

37·W = - 21·S

S = -37/21·W

W·S² = 5600

= W·(-37/21·W)² = 5600

W³ = 1803.94

W = 42.473 ft

S = √(5600/42.473)

S = 11.483 ft.