Question

Find the curved surface area (CSA) of the hemisphere if the volume is $2425\dfrac{1}{2}{m^3}$.

Answer 1

Hint – In order to solve this problem, firstly find the radius of the hemisphere with the help of the given volume then calculate its curved surface area by using the formula $2\pi {r^2}$.

Complete step by step answer:
We know the volume of the hemisphere of radius r is $\dfrac{2}{3}\pi {r^3}$. ……(1)
The given volume is $2425\dfrac{1}{2}{m^3}$. ……(2)
So, we can equate (1) and (2),
$\dfrac{2}{3}\pi {r^3} = 2425\dfrac{1}{2}$ ……(3)
As we know, ${\text{a}}\dfrac{{\text{b}}}{{\text{c}}}{\text{ = }}\dfrac{{{\text{ac + b}}}}{{\text{c}}}$ the same we will do with $2425\dfrac{1}{2}$.
Therefore,
$2425\dfrac{1}{2} = \dfrac{{2425(2) + 1}}{2} = \dfrac{{4851}}{2}$ …….(4)
From (3) and (4) we can say that,
$\dfrac{2}{3}\pi {r^3} = \dfrac{{4851}}{2}$ [Use $\pi = \dfrac{{22}}{7}$]
On solving it further to get the value of r we get,
$
{{\text{r}}^{\text{3}}{\text{ = }}\dfrac{4851 \times 7 \times 3}{2 \times 2 \times 22}} \\
  {{\text{r}}^{\text{3}}}{\text{ = 1157}}{\text{.625}} \\
  {{\text{r}}^{\text{3}}}{\text{ = }}\,{{\text{(10}}{\text{.5)}}^{\text{3}}} \\
  {\text{r = 10}}{\text{.5}} \\
$
Therefore the radius of the hemisphere is 10.5 m.
The curved surface area of the hemisphere is $2\pi {r^2}$. So, using the formula of CSA of hemisphere we get,
$
  CSA = 2\pi {r^2} = \,2 \times \dfrac{{22}}{7} \times {(10.5)^2} = \dfrac{{44}}{7} \times 110.25 \\
  CSA = \dfrac{{44}}{7} \times 110.25 = 693{m^2} \\
$
Hence the curved surface area of the hemisphere is 693 ${{\text{m}}^{\text{2}}}$.

Note – Whenever you face such types of problems with hemispheres or circles you just need to know one of the parameters that is radius. Since by knowing radius you can calculate their areas, volumes etc. Here we have given the volume and asked to find the CSA of the hemisphere. We have obtained the radius using the formula of volume then we have obtained the CSA of hemisphere using the formula of CSA. Doing this will take you to the right answer.