Question

Two cones have their heights in the ratio 1:3 and the radii of their bases in the ratio 3:1. Find the ratio of their volumes.

• A. 3:1
• B. 2:1
• C. 4:1
• D. 5:1

Hint:

When dealing with ratios of volumes or areas of similar geometric shapes, express the ratio of the formula variables and then substitute the given ratios. For example, for cones, $V \propto r^2h$, so $\frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^2 \left(\frac{h_1}{h_2}\right)$.

Answer 1

The Correct Option is A

Step 1: For a cone, volume scales with the square of the radius and directly with the height, so for two cones the volume ratio is simply \( \left(\frac{r_1}{r_2}\right)^2 \times \frac{h_1}{h_2} \).

Step 2: Substitute the given ratios directly: radius ratio \( \frac{r_1}{r_2} = \frac{3}{1} \) and height ratio \( \frac{h_1}{h_2} = \frac{1}{3} \).

Step 3: Compute: \( \left(\frac{3}{1}\right)^2 \times \frac{1}{3} = \frac{9}{1} \times \frac{1}{3} = \frac{9}{3} = 3 \).
\[ \boxed{3:1} \]