Question

The volume of a rectangular prism (shown below) is 48x^3+56x^2+16x Answer the following questions:

(1) What are the dimensions of the prism?
(2) If x = 2, use the polynomial 48x^3+56x^2+16x to find the volume of the prism.
(3) If x = 2, use the factors found in part a to calculate each dimension.
(4) Using the dimensions found in part c, calculate the volume. Show all work.

Answer 1

(a)

`Length = 8x\\Width = 3x + 2\\Height = 2x + 1`

(b)

`P(2) = 640`

(c)

`Length= 16`

`Width = 8`

`Height =5`

(d)

`Volume = 640`

Explanation:

Given

`P(x) = 48x^3 + 56x^2 + 16x`

Solving (a): The prism dimension

We have:

`P(x) = 48x^3 + 56x^2 + 16x`

Factor out 8x

`P(x) = 8x(6x^2 + 7x + 2)`

Expand 7x

`P(x) = 8x(6x^2 + 4x + 3x + 2)`

Factorize

`P(x) = 8x(2x(3x + 2) +1( 3x + 2))`

Factor out 3x + 2

`P(x) = 8x(3x + 2)(2x + 1)`

So, the dimensions are:

`Length = 8x\\Width = 3x + 2\\Height = 2x + 1`

Solving (b): The volume when
`x = 2`

We have:

`P(x) = 48x^3 + 56x^2 + 16x`

`P(2) = 48 * 2^3 + 56 * 2^2 + 16 * 2`

`P(2) = 640`

Solving (c): The dimensions when
`x = 2`

We have:

`Length = 8x\\Width = 3x + 2\\Height = 2x + 1`

Substitute 2 for x

`Length=8*2`

`Length= 16`

`Width = 3*2+2`

`Width = 8`

`Height = 2*2 + 1`

`Height =5`

So, we have:

`Length= 16`

`Width = 8`

`Height =5`

Solving (d), the volume in (c)

We have:

`Volume = Length * Width * Height`

`Volume = 16 * 8 * 5`

`Volume = 640`