Question

A manufacturer of shipping boxes has a box shaped like a cube. The side length is

(5a + 4b). What is the volume of the box in terms of a and b? Show your work.

Answer 1

The volume of a cube with side length equal to x, is
`V= x^(3)`,

thus, the volume of a cube shaped box, whose side length is (5a+4b) is :


`V= (5a+4b)^(3)`,

The volume is already expressed in terms of a and b, but we can expand the expression
`(5a+4b)^(3)`, as follows:


`(5a+4b)^(3) =(5a+4b)(5a+4b)^(2)= (5a+4b)[ (5a)^(2)+2(5a)(4b)+ (4b)^(2)]`


`=(5a+4b)[ 25a^(2)+40ab+ 16b^(2)]`


`=(5a)[ 25a^(2)+40ab+ 16b^(2)]+(4b)[ 25a^(2)+40ab+ 16b^(2)]`


`=125a^(3)+200a^(2) b+ 80ab^(2)+100a^(2)b+160ab^(2) + 64b^(3)`


`125a^(3)+300a^(2) b+240ab^(2) + 64b^(3)`


Answer:


`V= (5a+4b)^(3)`,

or


`125a^(3)+300a^(2) b+240ab^(2) + 64b^(3)`