Geometry

Solids

Medium

The ratio of radii of two right circular cylinders (A and B) is 2:3. The ratio of volumes of the cylinders A and B is 9:7, then what is the ratio of the heights of the cylinders A and B?

7:6

6:5

8:5

81:28

✅ Correct Option: 4

Given:

Ratio of radii (A : B) = 2 : 3
Ratio of volumes (A : B) = 9 : 7

The volume of a cylinder is $\pi r^2 h$

Let the radius of cylinder A be $r_A = 2k$ and the radius of cylinder B be $r_B = 3k$

The ratio of volumes can be written as:

$\dfrac{V_A}{V_B} = \dfrac{9}{7}$

$\dfrac{\pi r_A^2 h_A}{\pi r_B^2 h_B} = \dfrac{9}{7}$

$\dfrac{r_A^2 h_A}{r_B^2 h_B} = \dfrac{9}{7}$

Rearranging:

$\left(\dfrac{r_A}{r_B}\right)^2 \times \dfrac{h_A}{h_B} = \dfrac{9}{7}$

Substituting $\dfrac{r_A}{r_B} = \dfrac{2}{3}$ :

$\left(\dfrac{2}{3}\right)^2 \times \dfrac{h_A}{h_B} = \dfrac{9}{7}$

$\dfrac{4}{9} \times \dfrac{h_A}{h_B} = \dfrac{9}{7}$

Solving for the height ratio:

$\dfrac{h_A}{h_B} = \dfrac{9}{7} \times \dfrac{9}{4}$

$\dfrac{h_A}{h_B} = \dfrac{81}{28}$

Therefore, the ratio of heights (A : B) = 81 : 28

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The ratio of radii of two right circular cylinders (A and B) is 2:3. The ratio of volumes of the cylinders A and B is 9:7, then what is the ratio of the heights of the cylinders A and B?

Solution

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