Question

A shipping company offers various sized shipping boxes to its customers. Some of these boxes are cube-shaped, with equal height, width, and depth. As part of an upcoming sales promotion, the company will offer two cube-shaped boxes for the price of one.

a. Write an expression to represent the total volume of two different sized boxes as a sum of cubes if one of the boxes has sides with a length of 1 foot and the other has sides with a length of x feet.
b. Factor the sum of cubes.
c. Calculate the total volume of the two boxes if x = 3 feet.

Answer 1

A cube shaped box of side length equal to m foot has volume

`m*m*m= m^(3)` foot cubed.

a.
Let v1 be the volume of the box with side length 1 ft.
and v2 be the volume of the box with side length x ft.


`V1=1^(3)=1` (foot cubed)

`V2=x^(3)` (foot cubed)

Vtotal=
`V1+V2=1+x^(3)` feet cubed

b.

by the "sum of cubes" identity:


`x^(3)+1=(x+1)( x^(2) -x+1)`,

this identity can be derived by dividing
`x^(3)+1` by (x+1), by long division.

c. Vtotal when x=3 feet is:

Vtotal=
`V1+V2=1+3^(3)=1+27=28` (feet cubed)