A jewelry box with a square base is to be built with silver plated sides, nickel plated bottom and top, and a volume of 30 cm3. if nickel plating costs $1 per cm2 and silver plating costs $3 per cm2, find the dimensions of the box to minimize the cost of the materials

dimensions of the nickel plated square base: x*x = x^2

height: y

dimensions of the 4 silver plated sides: xy each

dimensions of the nickel plated top: x^2

Volume = 30cm^3 = yx^2 => y = 30 / x^2

Cost of the sides: 4 * xy * $3

Cost ot the top and the bottom: 2 * x^2 * $1

Total cost: 12xy + 2x^2

replace y by 30/x^2

=> cost = 12x * (30/x^2) + 2x^2 = 360 / x + 2x^2

Minimum cost => d [cost] / dx = 0 = - 360/x^2 + 4x =0

=> 90/x^2 - x =0

=> 90 - x^3 = 0

=> x^3 = 90

=> x = ∛(90) = 4.48

=> y = 30 / (4.48)^2 = 1.49

Answer: base: 4.48 cm* 4.48 cm; height: 1.49 cm