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