Question

Kartik has three solid objects — a cone, a hemisphere, and a cylinder. All three have the same base radius and the same height. He completely immerses each solid in a bucket full of water. What is the ratio of the volumes of the cylinder: cone: hemisphere?

• A. 1:2:3
• B. 3:1:2
• C. 2:1:3
• D. 1:1:1

Hint:

Always substitute \( h=r \) early to simplify the geometric expressions.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We must compare the volume formulas for the three shapes under the constraint that the base radius \( r \) and height \( h \) are equal. For a hemisphere, \( h = r \).

Key Formula or Approach:
Let the radius be \( r \) and the height be \( h=r \).
\( V_{cylinder} = \pi r^2 h = \pi r^2 (r) = \pi r^3 \)
\( V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (r) = \frac{1}{3} \pi r^3 \)
\( V_{hemisphere} = \frac{2}{3} \pi r^3 \)

Step 2: Detailed Explanation:
1. The volumes are \( \pi r^3 \), \( \frac{1}{3} \pi r^3 \), and \( \frac{2}{3} \pi r^3 \).
2. The ratio is \( \pi r^3 : \frac{1}{3} \pi r^3 : \frac{2}{3} \pi r^3 \).
3. Dividing throughout by \( \frac{1}{3} \pi r^3 \), we get: \( 3 : 1 : 2 \).

Step 3: Final Answer:
The ratio is 3:1:2.