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)