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:
[tex]\[
\text{Volume} = \text{Base Area} \times \text{Height}
\][/tex]
Plugging in the given values:
[tex]\[
24 = \text{Base Area} \times 2
\][/tex]
Solving for the base area:
[tex]\[
\text{Base Area} = \frac{24}{2} = 12 \text{ square inches}
\][/tex]
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 ([tex]\( L \)[/tex]) in terms of the width ([tex]\( W \)[/tex]) as:
[tex]\[
L = 3W
\][/tex]
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:
[tex]\[
\text{Base Area} = L \times W
\][/tex]
Substituting the given base area and the expression for [tex]\( L \)[/tex]:
[tex]\[
12 = (3W) \times W
\][/tex]
Simplifying, we get:
[tex]\[
12 = 3W^2
\][/tex]
4. Solve for the width [tex]\( W \)[/tex]:
[tex]\[
W^2 = \frac{12}{3} = 4
\][/tex]
Taking the square root of both sides:
[tex]\[
W = \sqrt{4} = -2 \text{ or } 2
\][/tex]
Since width cannot be negative, we take:
[tex]\[
W = 2 \text{ inches}
\][/tex]
5. Calculate the length [tex]\( L \)[/tex]:
Using the ratio 3:1:
[tex]\[
L = 3W = 3 \times 2 = 6 \text{ inches}
\][/tex]
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
