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: Fix the shared dimensions.
All three solids share the same base radius $r$ and the same height $h$. For a hemisphere the height equals the radius, so the common condition becomes $h = r$ for a fair comparison.
Step 2: Write the cylinder volume.
$V_{cyl} = \pi r^2 h$. Putting $h = r$ gives $V_{cyl} = \pi r^3$.
Step 3: Write the cone volume.
$V_{cone} = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi r^3$. A cone is always one-third of the cylinder on the same base and height, a handy fact to remember.
Step 4: Write the hemisphere volume.
$V_{hemi} = \dfrac{2}{3}\pi r^3$, which is half of a full sphere $\dfrac{4}{3}\pi r^3$.
Step 5: Form the ratio cylinder : cone : hemisphere.
$\pi r^3 : \dfrac{1}{3}\pi r^3 : \dfrac{2}{3}\pi r^3$. The common factor $\pi r^3$ cancels, leaving $1 : \dfrac{1}{3} : \dfrac{2}{3}$.
Step 6: Clear the fractions.
Multiply every term by 3 to get $3 : 1 : 2$. This matches option 2.
\[ \boxed{3 : 1 : 2} \]