​The ratio of the heights of a right circular cone and a right circular cylinder is 4 : 3 and the ratio of the radii of their bases is 7 : 3. If the volume of the cylinder is 810 $$\text{cm}^3$$​, then the volume (in $$\text{cm}^3$$) of the cone is:​

Correct option is D

Given:

Ratio of height of cone and cylinder = 4 : 3

Ratio of their radii of their bases = 7 : 3

Volume of cylinder = 810$$cm^3$$​​

Formula Used:

Volume of cone = $$\frac{1}{3}\pi r^2 h$$​​

Volume of cylinder =$$\pi r^2h$$​​

Solution:

Let the ratio of height of cone and cylinder be 4x and 3x respectively

And the ratio of radii of their bases be 7y and 3y

Then volume of cylinder = $$\pi (3y)^2 3x = \pi (27xy^2) = 810$$​​

​$$xy^2 = \frac{810}{27 \times \pi} = \frac{30}{\pi}$$​​

Volume of cone =$$\frac{1}{3}\pi (7y)^2 (4x) = \frac{1}{3}\pi (196xy^2)$$​​

​$$= \frac{1}{3}\times \pi \times 196 \times \frac{30}{\pi}$$​​

​$$= 1960cm^3$$​​