Question

Welolington is buiding a shed with a square base measuring x feet by x feet, and which is h feet high. He wants the volume of the shed to be 224 ft3. Concrete for the base costs $3 per ft2. The materials for each of the four sides cost $2 per ft2. The flat roof will cost $11 per ft2.

If Wellington wants to minimize cost, the the objective function in terms of x and h is:
C = .

The constraint equation in terms of x and h is:
224 = .


The dimensions which minimize cost are x = ft and h = ft.

The minimal cost to build the shed is $

Answer 1

224 =x^2 h
and C = 3x^2 +2*4x *h +11x^2=14x^2 +8xh=14x^2 +8x *(224/X^2)
C=14x^2 +1792/x
for max or min dC/dx=0 or 28x- 1792/x^2 =0 or 28x^3-1792=0
or x^3 =224

Answer 2

sorry for my wrong calculation
as 28x^3-1792=0
or x^3 -64 =0
or x^3=4^3
hence x=4
hence at x=4 the cost may be min

Answer 3

for checking we have d/dx(dC/dx) = 28 +2(1792/x^3)
and at x=4 d/dx(dC/dx) >0
hence at x=4 C is min
when x=4 h= 224/16 =14
and C=14(16) + 1792/16 =224 + 112 =336

Answer 4

oops again wrong calc for C
C=14(16) +1792/4 =224 + 448 =672 (neglect 336)