Application - Optimization

Question

A rectangular storage container with an open top is to have a volume of 28 cubic meters. The length of its base is twice the width. Material for the base costs 11 dollars per square meter. Material for the sides costs 6 dollars per square meter. Find the cost of materials for the cheapest such container.

Total cost =  (Round to the nearest penny and include monetary units. For example, if your answer is 1.095, enter $1.10 including the dollar sign and second decimal place.)

Answer 1

Cost function C(x) = 22x² + 36xh, where x is width of base and h is height.

2x²h = 28; h = 14/x²; C(x) = 22x² + 504/x;. C'(x) = 44x - 504/x² = 0; 44x³; x = (504/44)^1/3 = (126/11)^1/3;

C''(x) = 44 + 1008/x³; C''((126/11)^1/3) = 44 + 1008·11/126 = 132 > 0, so C(x) has min at x = (126/11)^1/3;

min C(x) = C((126/11)^1/3) = 22(126/11)^2/3 + 504(11/126)^1/3 = $335.37