The radius of the base and height of a cone are $3$ cm and $5$ cm respectively whereas the radius of the base and height of a cylinder are $2$ cm and $4$ cm respectively. The ratio of the volume of cone to that of the cylinder is

### Concept & Volume Ratios To find the ratio of their volumes, we simply use the standard volume formulas for a cone and a cylinder and divide them. $$ V_{\text{cone}} = \frac{1}{3}\pi \times r_1^2 \times h_1 $$ $$ V_{\text{cylinder}} = \pi \times r_2^2 \times h_2 $$ ### Step-by-Step Solution - **Given for Cone:** - Radius $r_1 = 3$ cm. - Height $h_1 = 5$ cm. - $V_{\text{cone}} = \frac{1}{3} \times \pi \times (3)^2 \times 5 = \frac{1}{3} \times \pi \times 9 \times 5 = 15\pi$. - **Given for Cylinder:** - Radius $r_2 = 2$ cm. - Height $h_2 = 4$ cm. - $V_{\text{cylinder}} = \pi \times (2)^2 \times 4 = \pi \times 4 \times 4 = 16\pi$. - **Calculation:** - Ratio = $\frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{15\pi}{16\pi} = \frac{15}{16}$. - The ratio is $15 : 16$. ### Exam Strategy & Shortcut Write the ratio as a single fraction initially to cancel out common terms like $\pi$ immediately: $\frac{(\frac{1}{3} \times 9 \times 5)}{(4 \times 4)} = \frac{15}{16}$. ### Common Pitfall Forgetting the $\frac{1}{3}$ factor in the volume of the cone, which would lead to an incorrect ratio of $45 : 16$. ### Final Answer Therefore, the correct answer is **15 : 16**.