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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.

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.

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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$=\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi}{R}^{3}}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^{3}}$

Therefore,

$\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi}{R}^{3}}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^{3}}=\frac{27}{8}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{R}^{3}}{{r}^{3}}=\frac{27}{8}\phantom{\rule{0ex}{0ex}}\Rightarrow {\left(\frac{R}{r}\right)}^{3}={\left(\frac{3}{2}\right)}^{3}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{R}{r}=\frac{3}{2}$

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

$=\frac{{R}^{2}}{{r}^{2}}\phantom{\rule{0ex}{0ex}}={\left(\frac{R}{r}\right)}^{2}\phantom{\rule{0ex}{0ex}}={\left(\frac{3}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}=\frac{9}{4}\phantom{\rule{0ex}{0ex}}=9:4$

Hence, Assertion (A) is false.

Reason (R): The given statement 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,

$\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi}{R}^{3}}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^{3}}=\frac{27}{8}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{R}^{3}}{{r}^{3}}=\frac{27}{8}\phantom{\rule{0ex}{0ex}}\Rightarrow {\left(\frac{R}{r}\right)}^{3}={\left(\frac{3}{2}\right)}^{3}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{R}{r}=\frac{3}{2}$

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

$=\frac{{R}^{2}}{{r}^{2}}\phantom{\rule{0ex}{0ex}}={\left(\frac{R}{r}\right)}^{2}\phantom{\rule{0ex}{0ex}}={\left(\frac{3}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}=\frac{9}{4}\phantom{\rule{0ex}{0ex}}=9:4$

Hence, Assertion (A) is false.

Reason (R): The given statement is true.

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