A rectangular box with a volume of 1088 ftcubed is to be constructed with a square base and top. The cost per square foot for the bottom is 15cents, for the top is 10cents, and for the sides is 2.5cents. What dimensions will minimize the cost?What are the dimensions of the box?A. The length of one side of the base is ____ ft?B. The hieght of the box is ____ft?
Accepted Solution
A:
Answer: x = 11,09 ft h = 8,85 ftStep-by-step explanation:Box with square base and topLet x b a side of the squareA₁ (area of base) A₁ = x² area of the top is equal to area of base A₂ = x²Cost of area A₁ C₁ = 15 * x²Cost of area A₂C₂ = 10 * x²Area of the side of height hA₃ = 2*π*x*h V = 1088 = x²*h h = 1088/x²A₃ = 2*π*x* (1088)/x² A₃ = 6832,64/xCost of A₃C₃ = 2,5*4 *6832,64/x C₃ = 68326,4/xTotal cost CC(x) = 15x² + 10x² + 68326,4/xTaking derivatives on both sides of the equationC´(x) = 30 x + 20 x - 68326,4/x²C´(x) = 50 x - 68326,4 /x² C´(x) = 0 50 x - 68326,4 /x² = 050x³ = 68326,4 x³ = 1366,53 x = 11,09 ft and the height h = 1088/x²h = 8,85 ft