Question

Help regarding this optimization problem to find dimensions?

In order to construct a box whose base length is 3 times the base width with volume of 50 ft^3, using materials to build the top and bottom that cost $ 10 per ft^2 and $ 6 per ft^2 for the sides. Determine the dimensions that will minimize the cost to build the box.

Answer 1

The box would have a length of `root3[180]`, a width of `[root3[180]]/3` and a height of `[150]/[root3[[180]^2]`

Explanation:

Let the box have a length of `x ft`, then the width will= `x/3ft` and let the height = `y`.

The area of the sides of the two sides of the box will be `2xy` and the area of the two ends of the box will equal `[2xy]/3` The area of the top and bottom of the box will equal `[2x^2]/3` [ all areas in `ft^2`]

From the question the areas of the sides will cost `6$` per square foot and the areas of the top and bottom of the box will cost `10$` per square foot , So we can say that the cost of the box......

`C=6[2xy+[2xy]/3]` + `[10[2x^2]/3]`.....= `[16xy + [20x^2]/3]`.......`[1]`

We are told that the volume of the box = `50 ft^3`, so `[x^2y]/3=50`, i.e,

`[y=[150]/x^2]`........`[2]` Substituting this value of `y` in .....`[1]`
`C=16x[150/x^2]+ 20x^2`, `C= [2400]/x + [20x^2]/3`, so we now have the cost of the box in terms of one variable.

Differentiating wrt `x` , `[dC]/dx =[ -[2400]/x^2 + [40x]/3]`, for max/min this expression must equal `0`.

When evaluated gives the answer `x= root3[180`. The second derivative, `[d^2C]/dx^2` = `4800/x^3+40/3`, which is positive when `x=root3[180]` indicating that this value of `x` will minimise the cost of the box.

The value of `y` can be found by substituting the value of `x` found into ........`[2]`. hope this was helpful.