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If the total surface area of a solid hemisphere is ${462c{m}^2}$, find its volume. (Take $\pi = \dfrac{{22}}{7}{\text{)}}$

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Hint: Equate the total surface area of the hemisphere to the formula and find out radius. Now, substitute the value of r in the Volume to get the answer.

Complete step by step answer:

Given, Total Surface area of solid hemisphere = $462cm^2$

Using the formula, Total Surface area of solid hemisphere = $3\pi {r^2}$

By comparing RHS of above two equations, we will find out the value for the radius of the hemisphere.

$\Rightarrow 3\pi {r^2} = 462 \Rightarrow {r^2} = \dfrac{{462}}{{3\pi }}$

Using $\pi {\text{ = }}\dfrac{{22}}{7}$

$\Rightarrow {r^2} = \dfrac{{462}}{{3 \times \dfrac{{22}}{7}}} = \dfrac{{462 \times 7}}{{3 \times 22}} = 49$

$\Rightarrow {\text{Radius, }}r = \sqrt {49} = 7cm$

As, we know that,

Volume of solid hemisphere, $V =\dfrac{2}{3}\pi {r^3}$

Now, substitute the value of r = 7cm and $\pi = \dfrac{{22}}{7}$

$\Rightarrow {\text{ V = }}\dfrac{2}{3} \times \dfrac{{22}}{7} \times {7^3} = \dfrac{{2156}}{3} = 718.67c{m^3}$

Therefore, the volume of the given solid hemisphere is $718.67c{m^3}$.

Note - These types of geometric problems can be easily solved with the use of some formulas related to a particular geometry asked in the problem and finding out some parameters like radius, height.