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: Since volume depends on radius squared and height as \(V \propto r^2 h\), we can find the ratio \(V_1:V_2\) just by tracking how each factor scales between the two cones.

Step 2: The radius ratio is \(r_1:r_2=3:1\), so squaring this scale factor contributes a multiplier of \(3^2=9\) toward \(V_1\) relative to \(V_2\). The height ratio is \(h_1:h_2=1:3\), which contributes a multiplier of \(\frac{1}{3}\) toward \(V_1\) relative to \(V_2\).

Step 3: Multiply the two scale factors together to get the overall volume ratio: \(9 \times \frac{1}{3} = 3\). So \(V_1:V_2 = 3:1\).
\[ \boxed{3:1} \]