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