Question

5. GEOMETRY The volume of a rectangular pyramid is one third the product of the area of its base and its height. Find

an expression for the volume of a rectangular pyramid whose base has an area of 3x2 + 12x + 9 square feet and
whose height is x + 3 feet.

Answer 1

x³ + 7x² + 15x + 9

Explanation:

Given that the volume (V) of a rectangular pyramid is

V =
`(1)/(3)` A h ( A is the area of base and h the height ), then

V =
`(1)/(3)` (3x² + 12x + 9)(x + 3) ← factor out 3 from A

=
`(1)/(3)` × 3(x² + 4x + 3)(x + 3)

= (x² + 4x + 3)(x + 3)

Each term in the second factor is multiplied by each term in the first factor, that is

x² (x + 3) + 4x(x + 3) + 3(x + 3) ← distribute parenthesis

= x³ + 3x² + 4x² + 12x + 3x + 9 ← collect like terms

= x³ + 7x² + 15x + 9

Thus the expression for the volume is

V = x³ + 7x² + 15x + 9