Question

A box witxh a height (x+5) cm has a square base with side x com. A second box with height (x+2) com has a square base with side (x+1) cm. If the two boxes have the same volume , find the value of x.

Answer 1

x=5.37 cm

Explanation:

we know that

The volume of the box is

`V=Bh`

where

B is the area of the base of the box

h is the height of the box

Box 1

we have that

The area of the base is

`B=x^2\ cm^2`

`h=(x+5)\ cm`

The volume of the Box 1

is equal to

`V_1=x^2(x+5)\ cm^3`

`V_1=(x^3+5x^2)\ cm^3`

Box 2

we have that

The area of the base is

`B=(x+1)^2\ cm^2`

`h=(x+2)\ cm`

The volume of the Box 2

is equal to

`V_2=[(x+1)^2(x+2)]\ cm^3`

`V_2=[(x^2+2x+1)(x+2)]\ cm^3`

`V_2=(x^3+2x^2+x+2x^2+4x+2)\ cm^3`

`V_2=(x^3+4x^2+5x+2)\ cm^3`

Equate the equation of Volume 1 to the equation of Volume 2

`(x^3+5x^2)=(x^3+4x^2+5x+2)`

`(5x^2)=(4x^2+5x+2)`

`5x^2-4x^2-5x-2=0`

`x^2-5x-2=0`

Solve the quadratic equation by graphing

using a graphing tool

The solution is x=5.37 cm

see the attached figure