A wise old troll wants to make a small hut. Roofing material costs five dollars per square foot and wall materials cost three dollars per square foot. According to ancient troll customs the floor must be square, but the height is not restricted.

(a) Express the cost of the hut in terms of its height h and the length x of the side of the square floor. ($)

(b) If the troll has only 960 dollars to spend, what is the biggest volume hut he can build? (ft^3)

a) Cost (h, x) = 12*x*h + 5*x²

b)

V = V (max) = 355.5 ft³

Dimensions of the hut:

x = 9.48 ft (side of the base square)

h = 3.95 ft (height of the hut)

Step-by-step explanation:

Let x be the side of the square of the base

h the height of the hut

Then the cost of the hut as a function of "x" and "h" is

Cost of the hut = cost of 4 sides + cost of roof

cost of side = 3 * x*h then for four sides cost is 12*x*h

cost of the roof = 5 * x²

Cost (h, x) = 12*x*h + 5*x²

If the troll has only 900 $

900 = 12xh + 5x² ⇒ 900 - 5x² = 12xh ⇒ (900-5x²) / 12x = h

And the volume of the hut is V = x²*h then

V (h) = x² * [ (900-5*x²) ]/12x

V (h) = x (900-5x²) / 12 ⇒ V (h) = (900*x - 5*x³) / 12

Taking derivatives (both sides of the equation):

V' (h) = (900 - 10 * x²) / 12 V' (h) = 0

900 - 10*x² = 0 ⇒ x² = 90 x = √90

x = 9.48 ft

And h

h = (900-5x²) / 12x ⇒ h = [900 - 90 (5) ]/12*x ⇒ h = 450/113,76

h = 3.95 ft

And finally the volume of the hut is:

V (max) = x²*h ⇒ V (max) = 90*3.95

V (max) = 355.5 ft³