Question

figure below shows a rectangular prism. one base of the prism is shaded.
(a) write an expression to find the volume of the prism using its length, width, and height. then simplify.
volume of prism: \\(\square \times \square \times \square = \square\\) cubic units
(b) find the area of the shaded base. then, using the area of the shaded base, write an expression to find the volume of the prism and simplify.
area of shaded base: \\(\square \times \square = \square\\) square units
volume of prism: \\(\square \times \square = \square\\) cubic units

Explanation:

Part (a)

Step1: Recall volume formula for rectangular prism

The volume \( V \) of a rectangular prism is given by the formula \( V = \text{length} \times \text{width} \times \text{height} \). From the figure, length \( = 9 \), width \( = 2 \), height \( = 8 \).

Step2: Substitute values into the formula

Substitute length \( = 9 \), width \( = 2 \), height \( = 8 \) into the formula: \( 9 \times 2 \times 8 \).

Step3: Simplify the expression

First, multiply \( 9 \times 2 = 18 \), then \( 18 \times 8 = 144 \). So \( 9 \times 2 \times 8 = 144 \).

Step1: Recall area formula for rectangle

The shaded base is a rectangle with length \( = 9 \) and width \( = 2 \). The area \( A \) of a rectangle is \( A=\text{length} \times \text{width} \).

Step2: Substitute values into the formula

Substitute length \( = 9 \) and width \( = 2 \) into the formula: \( 9 \times 2 = 18 \).

Step1: Recall volume formula using base area

The volume \( V \) of a prism is given by \( V=\text{Area of base} \times \text{height} \). The area of the base is \( 18 \) (from above) and the height of the prism is \( 8 \).

Step2: Substitute values into the formula

Substitute area of base \( = 18 \) and height \( = 8 \) into the formula: \( 18 \times 8 \).

Step3: Simplify the expression

\( 18 \times 8 = 144 \).

Answer:

Volume of prism: \( 9 \times 2 \times 8 = 144 \) cubic units

Part (b)

Area of shaded base