Question

A closed box with a square bottom is three times high as it is wide. a) Express the surface area of the box in terms of its width. b) Express the volume of the box in terms of its width. c) Express the surface area in terms of the volume. d) If the box has a volume of 24 m³, what is its surface area?

Answer 1

a)
`S(s) = 14s^2`

b)
`V(s) = 3s^3`

c)
`S(s) = (14V(s))/(3s)`

d) 56 square meter

Explanation:

We are given the following in the question:

A closed box with a square bottom is three times high as it is wide.

Let s be the side of square and h be the height.

`h = 3s`

a) Surface area of box

`2(lb + bh + hl)`

where l is the length, b is the breadth and h is the height.

Putting values:

`S = 2(s^2 + sh +sh)\\S = 2(s^2 + 3s^2 + 3s^2)\\S(s) = 14s^2`

b) Volume of box

`l* b * h`

where l is the length, b is the breadth and h is the height.

Putting values:

`V = s* s* h\\V= s* s* 3s\\V(s) = 3s^3`

c) Surface area in terms of volume

`S(s) = 14s^2 = (14V(s))/(3s)`

d) Surface area

Volume = 24 m³

`V(s) = 24\\3s^3 = 24\\s^3 = 3\\s = 2`

`S(2) = 14(2)^2 = 56\text{ square meter}`