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 = $\frac{4}{3}\mathrm{\pi }{R}^3$
Surface area of a sphere = $4\mathrm{\pi }{R}^2$

(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.

Answer 1

(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$=\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}$
Therefore,
$$\begin{gathered} \frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}=\frac{27}{8} \\ \Rightarrow \frac{R^3}{r^3}=\frac{27}{8} \\ \Rightarrow {\left(\frac{R}{r}\right)}^3={\left(\frac{3}{2}\right)}^3 \\ \Rightarrow \frac{R}{r}=\frac{3}{2} \end{gathered}$$
Hence, the ratio of their surface areas$=\frac{4\mathrm{\pi }{R}^2}{4{\mathrm{\pi r}}^2}$
$$\begin{gathered} =\frac{R^2}{r^2} \\ ={\left(\frac{R}{r}\right)}^2 \\ ={\left(\frac{3}{2}\right)}^2 \\ =\frac{9}{4} \\ =9:4 \end{gathered}$$
Hence, Assertion (A) is false.

Reason (R): The given statement is true.