Question

use the picture to answer the questions.

image of a rectangular prism with length ( x^2 + 3x ), width ( x - 2 ), and height ( x - 3 )

1. write and simplify an expression for the volume of the rectangular prism.

1. calculate the volume if ( x = 4 ) feet.

Explanation:

Question 1

Step1: Recall volume formula

The volume \( V \) of a rectangular prism is given by the product of its length, width, and height. So, \( V = l \times w \times h \), where \( l = x^2 + 3x \), \( w = x - 2 \), and \( h = x - 3 \).
\[ V=(x^{2}+3x)(x - 2)(x - 3) \]

Step2: Factor \( x^2 + 3x \)

Factor out \( x \) from \( x^2 + 3x \), we get \( x(x + 3) \). So the volume becomes:
\[ V = x(x + 3)(x - 2)(x - 3) \]

Step3: Rearrange and multiply

Rearrange the factors: \( V = x(x - 2)(x + 3)(x - 3) \). Notice that \( (x + 3)(x - 3) \) is a difference of squares, \( (x + 3)(x - 3)=x^{2}-9 \), and \( x(x - 2)=x^{2}-2x \). Now multiply \( (x^{2}-2x)(x^{2}-9) \):
\[

$$\begin{align*} (x^{2}-2x)(x^{2}-9)&=x^{2}(x^{2}-9)-2x(x^{2}-9)\\ &=x^{4}-9x^{2}-2x^{3}+18x\\ &=x^{4}-2x^{3}-9x^{2}+18x \end{align*}$$

\]

Step1: Substitute \( x = 4 \) into the volume formula

We can use the factored form or the expanded form. Let's use the factored form \( V = x(x + 3)(x - 2)(x - 3) \). Substitute \( x = 4 \):
\[

$$\begin{align*} V&=4\times(4 + 3)\times(4 - 2)\times(4 - 3)\\ &=4\times7\times2\times1 \end{align*}$$

\]

Step2: Calculate the product

Multiply the numbers: \( 4\times7 = 28 \), \( 28\times2 = 56 \), \( 56\times1 = 56 \).

Answer:

The volume of the rectangular prism is \( \boldsymbol{x^{4}-2x^{3}-9x^{2}+18x} \) (or also can be written as \( x(x - 2)(x + 3)(x - 3) \) before full expansion).

Question 2