The radii of the two spheres are in the ratio 4 : 5. Their volume will be in the ratio:

Correct option is D

Given:

The ratio of the radii of two spheres = 4 : 5.

Formula Used:

Volume of a sphere:

V$$= \frac{4}{3} \pi r^3$$​

Thus, the ratio of volumes = (ratio of radii)$$^3.$$​​

Solution:

Let the radii of the two spheres be 4k and 5k (since the ratio is 4 : 5).
Volume of the first sphere $$( V_1):$$​​
$$V_1 = \frac{4}{3} \pi (4k)^3 = \frac{4}{3} \pi (64k^3) = \frac{256}{3} \pi k^3$$​​

Volume of the second sphere $$( V_2):$$​​

​$$V_2 = \frac{4}{3} \pi (5k)^3 = \frac{4}{3} \pi (125k^3) = \frac{500}{3} \pi k^3$$​​

$$\frac{V_1}{V_2} = \frac{\frac{256}{3} \pi k^3}{\frac{500}{3} \pi k^3} = \frac{256}{500} = \frac{64}{125}$$​​
Alternate Method :
Since volume $$\propto (\text{radius})^3,$$​​
Volume ratio = $$(4 : 5)^3 = 4^3 : 5^3 = 64 : 125$$​
The volumes are in the ratio 64 : 125.