1 Marks Question

Question 11 Mark

Volume and surface area of a solid hemisphere are numerically equal.
What is the diameter of hemisphere?

Answer

We know that volume of a solid hemisphere is given by
$
V=\frac{2}{3} \pi r^3
$
Also, surface area of a solid hemisphere is given by
$
S=3 \pi r^2
$
Where, $r$ is the radius of the solid hemisphere According to the question,
Volume and surface area of a solid hemisphere are numerically equal
$
\begin{aligned}
\frac{2}{3} \pi r^3 & =3 \pi r^2 \\
\Rightarrow 2 r & =9
\end{aligned}
$
We know that $2 r=$ diameter
$\Rightarrow$ diameter $=9$ units
Hence, the diameter of the solid hemisphere is 9 units.

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Question 21 Mark

If the radii of two spheres are in the ratio $2: 3$, then the ratio of their respective volumes is $.......$

Answer

$\frac{r_1}{r_2}=\frac{2}{3}$
$\frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3}$
$\Rightarrow\left(\frac{r_1}{r_2}\right)^3$
$\Rightarrow\left(\frac{2}{3}\right)^3=\frac{8}{27}$
$\text { Ratio of volumes }=8: 27$

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