Question

the volume of a rectangular box is $2x(2x + 4)(2x - 2)$. (drawing is not to scale.)

image of a rectangular box with length $2x + 4$, width $2x$, height $2x - 2$

which statement about the volume of the box is true?

a. the volume is the product of the area of the base, $2x(2x + 4)$, and the height, $2x - 2$.

b. the volume is the product of the length, $2x + 4$, and the width, $2x$.

c. the volume does not depend on the height, $2x - 2$.

d. the volume is the sum of the length, $2x + 4$, the width, $2x$, and the height, $2x - 2$.

Explanation:

Step1: Recall Volume of Rectangular Box

The volume \( V \) of a rectangular box is given by the formula \( V=\text{length} \times \text{width} \times \text{height} \). Also, the area of the base (which is a rectangle) is \( \text{length} \times \text{width} \), so \( V = \text{Area of base} \times \text{height} \).

Step2: Analyze Each Option

• Option A: The area of the base is \( \text{length} \times \text{width}=2x(2x + 4) \), and height is \( 2x-2 \). So volume \( = 2x(2x + 4)\times(2x - 2) \), which matches the given volume formula. This seems correct.
• Option B: The product of length and width is the area of the base, not the volume. Volume requires multiplying by height too. So this is wrong.
• Option C: Volume of a rectangular box depends on height (since \( V=\text{length}\times\text{width}\times\text{height} \)), so it does depend on \( 2x - 2 \). This is wrong.
• Option D: Volume of a rectangular box is the product, not the sum, of length, width, and height. So this is wrong.

Answer:

A. The volume is the product of the area of the base, \( 2x(2x + 4) \), and the height, \( 2x - 2 \).