Question

Optimization problem?

A jewelry box with a square base is to be built with silver plated sides, nickel plated bottom and top, and a volume of `32cm^3`. If nickel plating costs $1 per `cm^2` and silver plating costs $2 per `cm^2`, find the dimensions of the box to minimize the cost of the materials. (Round your answers to two decimal places.)

The box which minimizes the cost of materials has a square base of side length `?` cm and a height of `?` cm

Answer 1

`h=5.04cm` and `x=2.52cm`

Explanation:

let the side of the base `= x , cm` and the height `= h , cm`

So, if V is the volume then it is given by `V=x^2h` or `x^2h = 32`

but the Cost ($) , `C = 4x^2+4xh`

since we have two variable's then from `h=32/x^2`

`C=4x^2+4x(32/x^2)` so `C=4x^2+128/x`

then `C'=8x-128/x^2` but for minimum cost we put `C'=0`

then `128/x^2=8x` or `x^3=16or x=2.52cm` then from `h=32/x^2`

then `h=32/(16^(1/3))^2` then `h=5.04cm`

therefore the dimensions that minimize the cost are

`h=5.04cm` and `x=2.52cm`