Answered

The volume of a rectangular prism is 24 cubic inches. The prism is 2 inches high, and the ratio of the
length of a base to the width of the base is 3:1. What are the dimensions of the base of the prism?
Complete the explanation of your reasoning.
The volume is 24 cubic inches, and the height is 2 inches. So the area of the base is
inches. Since the ratio length:width = 3:1, the length is
inches.
square
inches, and the width is

Verified Answer :

GinnyAnswer

Let's solve this step-by-step:

1. Determine the area of the base:
We are given that the volume of the rectangular prism is 24 cubic inches, and the height is 2 inches.
The formula for the volume of a rectangular prism is given by:
\(\[ \text{Volume} = \text{Base Area} \times \text{Height} \]\)
Plugging in the given values:
\(\[ 24 = \text{Base Area} \times 2 \]\)
Solving for the base area:
\(\[ \text{Base Area} = \frac{24}{2} = 12 \text{ square inches} \]\)

2. Utilize the given ratio to express length in terms of width:
We know that the ratio of the length to the width of the base is 3:1. Therefore, we can express the length (\(\( L \)\)) in terms of the width (\(\( W \)\)) as:
\(\[ L = 3W \]\)

3. Set up the equation for the area of the base:
Since the area of the base is given by the product of its length and width, we can write:
\(\[ \text{Base Area} = L \times W \]\)
Substituting the given base area and the expression for \(\( L \)\):
\(\[ 12 = (3W) \times W \]\)
Simplifying, we get:
\(\[ 12 = 3W^2 \]\)

4. Solve for the width \(\( W \)\):
\(\[ W^2 = \frac{12}{3} = 4 \]\)
Taking the square root of both sides:
\(\[ W = \sqrt{4} = -2 \text{ or } 2 \]\)
Since width cannot be negative, we take:
\(\[ W = 2 \text{ inches} \]\)

5. Calculate the length \(\( L \)\):
Using the ratio 3:1:
\(\[ L = 3W = 3 \times 2 = 6 \text{ inches} \]\)

Therefore:
- The base area is 12 square inches.
- The length is 6 inches.
- The width is 2 inches.

In summary, the dimensions of the base of the prism are:
- Base area: 12 square inches
- Length: 6 inches
- Width: 2 inches

Related Questions