Mathematics: A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs $0.34 per square foot, the material for the sides costs $0.05 per square foot, and the material for the top costs $0.16 per square foot, determine the dimensions of the box that can be constructed at minimum cost.

-178Step-by-step explanation:go left of the 0 to -212 then go right toward the 0 34 spaces to -178...

A rectangular box is to have a square base and a

Ответ:

youngchapo813p8d9u1

29.04.2021

Mathematics

The dimensions of the box so that total costs are minimum are a side length of 2 feet and a height of 5 feet.

Step-by-step explanation:

Geometrically speaking, the volume of the rectangular box (), in cubic feet, is represented by this formula:

$V = l^{2}\cdot h$ (1)

Where:

- Side length of the box, in feet.

- Height of the box, in feet.

In addition, the total cost of the box (), in monetary units, is defined by this formula:

$C = (c_{b}+c_{t})\cdot l^{2} + 4\cdot c_{s}\cdot l\cdot h$ (2)

Where:

$c_{b}$ - Unit cost of the base of the box, in monetary units per square foot.

$c_{t}$ - Unit cost of the top of the box, in monetary units per square foot.

$c_{s}$ - Unit cost of the side of the box, in monetary units per square foot.

By (1), we clear into the expression:

$h = \frac{V}{l^{2}}$

And we expand (2) and simplify the resulting expression:

$C = (c_{b}+c_{t})\cdot l^{2}+4\cdot c_{s}\cdot \left(\frac{V}{l} \right)$ (3)

If we know that $c_{b} = 0.34\,\frac{m.u.}{ft^{2}}$ , $c_{s} = 0.05\,\frac{m.u.}{ft^{2}}$ , $c_{t} = 0.16\,\frac{m.u.}{ft^{2}}$ and $V = 40\,ft^{3}$ , then we have the resulting expression and find the critical values associated with the side length of the base: