Question

Kritika has three solid objects cone, hemisphere, cylinder. All have same base radius and height. Then the ratio of volume of cylinder : cone : hemisphere is

• A. \(1:2:3\)
• B. \(1:1:1\)
• C. \(3:1:2\)
• D. \(2:1:3\)

Hint:

Remember: \[ \text{Cylinder : Cone} = 3:1 \] for equal radius and height.

Answer 1

The Correct Option is C

Step 1: Note the shared measurements.
All three solids share the same base radius $r$ and the same height. For a hemisphere the natural height is its radius, so we take $h=r$ throughout.
Step 2: Write the cylinder volume.
\[ V_{cyl}=\pi r^2 h = \pi r^2 (r)=\pi r^3 \]
Step 3: Write the cone volume.
\[ V_{cone}=\frac{1}{3}\pi r^2 h = \frac{1}{3}\pi r^3 \]
Step 4: Write the hemisphere volume.
\[ V_{hemi}=\frac{2}{3}\pi r^3 \]
Step 5: Stack them as a ratio.
\[ \pi r^3 : \frac{1}{3}\pi r^3 : \frac{2}{3}\pi r^3 \] Cancel the common $\pi r^3$ to get $1:\frac13:\frac23$.
Step 6: Clear the fractions.
Multiply every part by $3$: \[ 3:1:2 \] So cylinder : cone : hemisphere is $3:1:2$.
\[ \boxed{3:1:2} \]