Question

Assertion: If the radius of a sphere is doubled then the ratio of the volume of the first sphere to that of the second is 1 : 8.
Reason: A cone and a hemisphere have equal bases and equal volumes. The ratio of their heights is 1 : 2.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true and Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Answer 1

(c) Assertion is true and Reason is false.
Assertion (A):
Let r be the radius of the sphere. On doubling it, it becomes 2r.
$$\begin{gathered} \mathrm{Ratio}\mathrm{of}\mathrm{the}\mathrm{volumes}=\frac{4}{3}\mathrm{\pi }{r}^3:\frac{4}{3}\mathrm{\pi }{\left(2r\right)}^3 \\ =1:8 \end{gathered}$$

Reason (R):
It is given that the cone and the hemisphere have equal bases and equal volumes.


$$\begin{gathered} \mathrm{Let}\mathrm{their}\mathrm{radii}\mathrm{be}r\mathrm{and}\mathrm{let}h\mathrm{be}\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone}. \\ \therefore \frac{\mathrm{Volume}\mathrm{of}\mathrm{cone}}{\mathrm{Volume}\mathrm{of}\mathrm{hemisphere}}=\frac{{\displaystyle \frac{1}{3}}\mathrm{\pi }{r}^{\mathit{2}}h}{{\displaystyle \frac{2}{3}}\mathrm{\pi }{r}^3} \\ \mathrm{Since}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{cone}=\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{hemisphere},\mathrm{we}\mathrm{have}: \\ \frac{1}{3}\pi r \end{gathered}$$²h = 23πr³ ⇒h = 2r ⇒h:r = 2:1

Hence, reason (R) is false.

Assertion (A) is true but reason (R) is clearly false and (R) is not the explanation for (A).