Question

A container manufacturer plans to make rectangular boxes whose bottom and top measure 3x by 4x. The container must contain 48 cm^3. The top and the bottom will cost $3.50 per square centimeter, while the four sides will cost $4.40 per square centimeter.

What should the height of the container be so as to minimize cost? Round your answer to the nearest hundredth.

Answer 1

The height of the container that minimize cost is 3.08 cm

Explanation:

Let

h ---> the height of the container

we know that

The volume of the box is equal to

`V=(3x)(4x)h`

`V=12x^2h`

`V=48\ cm^3`

substitute

`48=12x^2h`

`h=(4)/(x^2)`

The function cost is equal to

`C=3.50(2)(12x^2)+4.40(14x)(h)`

`C=84x^2+61.6x(4)/(x^2)`

`C=84x^2+(246.4)/(x)`

To find out the minimum cost determine the first derivative

`(dC)/(dx)=168x-(246.4)/(x^2)`

equate to zero

`168x=(246.4)/(x^2)\\x^3=1.4667\\x=1.14\ cm`

Find the height of the container

`h=(4)/(x^2)`

substitute the value of x

`h=(4)/(1.14^2)=3.08\ cm`