Question

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

Answer 1

Answer:

(a)

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

(b)

$P(2) = 640$

(c)

$Length= 16$

$Width = 8$

$Height =5$

(d)

$Volume = 640$

Step-by-step 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:

$Volume = Length * Width * Height$

$Volume = 16 * 8 * 5$

$Volume = 640$