Question

A rectangular box with a volume of 272ft^3 is to be constructed with a square base and top. The cost per square foot for the bottom is 20 cents, for the top is 10 cnets, and the sides 1.5 cents. What dimensions will minimize the cost?

What are the dimensions of the box?
The length of one side of the base is ____ ft?
The hieght of the box is ____ft?

Answer 1

The dimensions of the box is 3 ft by 3 ft by 30.22 ft.

The length of one side of the base of the given box is 3 ft.

The height of the box is 30.22 ft.

Explanation:

Given that, a rectangular box with volume of 272 cubic ft.

Assume height of the box be h and the length of one side of the square base of the box is x.

Area of the base is =
`(x* x)`

`=x^2`

The volume of the box is = area of the base × height

`=x^2h`

Therefore,

`x^2h=272`

`\Rightarrow h=(272)/(x^2)`

The cost per square foot for bottom is 20 cent.

The cost to construct of the bottom of the box is

=area of the bottom ×20

`=20x^2` cents

The cost per square foot for top is 10 cent.

The cost to construct of the top of the box is

=area of the top ×10

`=10x^2` cents

The cost per square foot for side is 1.5 cent.

The cost to construct of the sides of the box is

=area of the side ×1.5

`=4xh* 1.5` cents

`=6xh` cents

Total cost =
`(20x^2+10x^2+6xh)`

`=30x^2+6xh`

Let

C
`=30x^2+6xh`

Putting the value of h

`C=30x^2+6x* (272)/(x^2)`

`\Rightarrow C=30x^2+(1632)/(x)`

Differentiating with respect to x

`C'=60x-(1632)/(x^2)`

Again differentiating with respect to x

`C''=60+(3264)/(x^3)`

Now set C'=0

`60x-(1632)/(x^2)=0`

`\Rightarrow 60x=(1632)/(x^2)`

`\Rightarrow x^3=(1632)/(60)`

`\Rightarrow x\approx 3`

Now
`C''|_(x=3)=60+(3264)/(3^3)>0`

Since at x=3 , C''>0. So at x=3, C has a minimum value.

The length of one side of the base of the box is 3 ft.

The height of the box is
`=(272)/(3^2)`

=30.22 ft.

The dimensions of the box is 3 ft by 3 ft by 30.22 ft.