Question

A rectangular solid has a square base and altitude of 7. If the volume of the solid is 252, then the perimeter of the square base is

• A. 9
• B. 24
• C. 28
• D. 36
• E. 49

Hint:

For "working backward" problems in geometry, list the formulas you know and identify the unknown you need. Start with the given information (like volume) and solve for the intermediate value (like base area or side length) that you need to get to the final answer.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a solid geometry problem. We are given the volume, shape of the base, and height of a rectangular solid (a prism), and we need to work backward to find the perimeter of the base.
Step 2: Key Formula or Approach:
1. Volume of a rectangular solid: \(V = (\text{Area of Base}) \times (\text{Altitude})\).
2. For a square base with side length \(s\), the Area = \(s^2\).
3. Perimeter of a square base = \(4s\).
Step 3: Detailed Explanation:
1. Use the volume formula to find the area of the base.
Let \(A_{base}\) be the area of the square base.
We are given \(V = 252\) and Altitude (height) \(h = 7\).
\[ 252 = A_{base} \times 7 \]
Divide by 7 to solve for the area of the base:
\[ A_{base} = \frac{252}{7} = 36 \]
2. Find the side length of the square base.
The area of a square is \(s^2\).
\[ s^2 = 36 \]
Take the square root of both sides:
\[ s = \sqrt{36} = 6 \]
The side length of the square base is 6.
3. Calculate the perimeter of the base.
The perimeter of a square is \(4s\).
\[ \text{Perimeter} = 4 \times 6 = 24 \]
Step 4: Final Answer:
The perimeter of the square base is 24.