Question

Find the curved surface area and total surface area of a hollow hemisphere whose outer and inner radii are 4.3 cm and 2.1 cm respectively.
$
  \left( a \right)36.98\pi c{m^2},59.26\pi c{m^2} \\
  \left( b \right)48.1\pi c{m^2},58.22\pi c{m^2} \\
  \left( c \right)36.98\pi c{m^2},59.88\pi c{m^2} \\
  \left( d \right)49.1\pi c{m^2},5.32\pi c{m^2} \\
$

Answer 1

Hint: In this question, we use the formula of curved surface area and total surface area of hemisphere. Curved surface area of hemisphere $ = 2\pi {r^2}$ and Total surface area of hemisphere $ = 3\pi {r^2}$

Complete Step-by-Step solution:
We have a hollow hemisphere whose inner radius, r= 2.1cm and outer radius, R=4.3cm.
Now, the curved surface area of a hollow hemisphere is the same as the curved surface area of the outer hemisphere.
Curved surface area of a hollow hemisphere $ = 2\pi {R^2}$
R is the radius of outer hemisphere, R=4.3cm
\[
   \Rightarrow C.S.A{\text{ of hollow hemisphere}} = 2\pi \times {\left( {4.3} \right)^2} \\
   \Rightarrow C.S.A{\text{ of hollow hemisphere}} = 2 \times \pi \times {\left( {4.3} \right)^2} \\
   \Rightarrow C.S.A{\text{ of hollow hemisphere}} = 2 \times \pi \times 18.49 \\
   \Rightarrow C.S.A{\text{ of hollow hemisphere}} = 36.98\pi c{m^2} \\
 \]
Now,
Total surface area of a hollow hemisphere= curved surface area of outer hemisphere + curved surface area of inner hemisphere +area of ring
$
   \Rightarrow T.S.A{\text{ of hollow hemisphere}} = 2\pi {R^2} + 2\pi {r^2} + \pi \left( {{R^2} - {r^2}} \right) \\
   \Rightarrow T.S.A{\text{ of hollow hemisphere}} = 2\pi {R^2} + 2\pi {r^2} + \pi {R^2} - \pi {r^2} \\
   \Rightarrow T.S.A{\text{ of hollow hemisphere}} = 3\pi {R^2} + \pi {r^2} \\
 $
Take common $\pi $
$ \Rightarrow T.S.A{\text{ of hollow hemisphere}} = \pi \left( {3{R^2} + {r^2}} \right)$
We know, R=4.3cm and r=2.1cm
\[
   \Rightarrow T.S.A{\text{ of hollow hemisphere}} = \pi \left( {3 \times {{\left( {4.3} \right)}^2} + {{\left( {2.1} \right)}^2}} \right) \\
   \Rightarrow T.S.A{\text{ of hollow hemisphere}} = \pi \left( {3 \times 18.49 + 4.41} \right) \\
   \Rightarrow T.S.A{\text{ of hollow hemisphere}} = \pi \left( {55.47 + 4.41} \right) \\
   \Rightarrow T.S.A{\text{ of hollow hemisphere}} = 59.88\pi c{m^2} \\
\]
So, the correct option is (c).

Note: In such types of problems curved surface area of hollow hemisphere is same as the curved surface area of hemisphere and total surface area of hollow hemisphere is equal to the sum of curved surface area of outer and inner hemisphere and also area of ring.