JIPMAT 2023 (QA) - <p>The radii of two cones are in the ratio 2:3 and their volumes are in the ratio 1:3. Then the ratio of their heights is:</p> | PYQs + Solutions

JIPMAT 2023

Geometry

Solids

Conceptual

The radii of two cones are in the ratio 2:3 and their volumes are in the ratio 1:3. Then the ratio of their heights is:

3:2

1:2

3:4

2:3

Correct Option: 3

1) Let's use the formula for volume of a cone:

V = \frac{1}{3}πr^2h

, where

r

is radius and

h

is height

2) Given:

\newline

- Ratio of radii

(r_1:r_2) = 2:3

\newline

- Ratio of volumes

(V_1:V_2) = 1:3

3) Let's express volume ratio using formula:

\frac{V_1}{V_2} = \frac{\frac{1}{3}πr_1^2h_1}{\frac{1}{3}πr_2^2h_2} = \frac{r_1^2h_1}{r_2^2h_2} = \frac{1}{3}

4) Now substitute radius ratio:

\frac{r_1}{r_2} = \frac{2}{3}

, then

\frac{r_1^2}{r_2^2} = \frac{4}{9}

5) Put this in volume ratio equation:

\frac{4}{9} × \frac{h_1}{h_2} = \frac{1}{3}

6) Solve for height ratio:

\frac{h_1}{h_2} = \frac{1}{3} × \frac{9}{4} = \frac{3}{4}

Therefore, the ratio of heights

(h_1:h_2) = 3:4

The answer is

3:4