 # Assertion (A) If the volumes of two spheres are in the ratio 27 : 8,

Question:

Assertion (A)
If the volumes of two spheres are in the ratio 27 : 8, then their surface areas are in the ratio 3 : 2.

Reason (R)

Volume of a sphere $=\frac{4}{3} \pi R^{3}$

Surface area of a sphere $=4 \pi R^{2}$

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Solution:

(d) Assertion (A) is false and Reason (R) is true.
Assertion (A):
Let R and r be the radii of the two spheres.

Then, ratio of their volumes $=\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi r^{3}}$

Therefore,

$\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi r^{3}}=\frac{27}{8}$

$\Rightarrow \frac{R^{3}}{r^{3}}=\frac{27}{8}$

$\Rightarrow\left(\frac{R}{r}\right)^{3}=\left(\frac{3}{2}\right)^{3}$

$\Rightarrow \frac{R}{r}=\frac{3}{2}$

Hence, the ratio of their surface areas $=\frac{4 \pi R^{2}}{4 \pi r^{2}}$

$=\frac{R^{2}}{r^{2}}$

$=\left(\frac{R}{r}\right)^{2}$

$=\left(\frac{3}{2}\right)^{2}$

$=\frac{9}{4}$

$=9: 4$

Hence, Assertion (A) is false.

Reason (R): The given statement is true.