Question

Draw an isometric sketch for the shape obtained by joining two hemispheres of radius $ 5cm $ and then find the surface area and volume of the shape obtained.
A. $ 219.35\,c{m^2}\,and\,\,423.65\,c{m^3} $
B. $ 256.36\,c{m^2}\,and\,\,128.45\,c{m^2} $
C. $ 314.15\,c{m^2}\,\,and\,\,450.38\,c{m^3} $
D. $ 314.15\,c{m^2}\,\,and\,\,523.59\,c{m^3} $

Answer 1

Hint: We know that if we cut a sphere into two equal parts then two hemispheres will be formed of the same radius as the sphere. Hence, on joining two hemispheres of the same radius a sphere will be obtained of the same radius and its volume and surface area can be calculated by using formulas of mensuration.
Volume of sphere = $ \dfrac{4}{3}\pi {r^3}, $ Surface area of sphere = $ 4\pi {r^2}, $ where ‘r’ radius of sphere.

Complete step-by-step answer:

Radius of given hemisphere = $ 5cm $
As, sphere is obtained by adjoining two hemispheres. Therefore its radius will also be the same as the radius of the hemisphere.
Radius of sphere = $ 5cm $
Also, we know that volume of sphere is given as = $ \dfrac{4}{3}\pi {r^3}, $
Substituting value of ‘r’ in above volume formula we have
 $
\Rightarrow Volume(V) = \dfrac{4}{3} \times \dfrac{{22}}{7} \times {(5)^3} \\
\Rightarrow Volume(V) = \dfrac{4}{3} \times \dfrac{{22}}{7} \times 5 \times 5 \times 5 \\
\Rightarrow Volume(V) = \dfrac{4}{3} \times \dfrac{{22}}{7} \times 125 \\
\Rightarrow Volume(V) = \dfrac{{11000}}{{21}} \\
\Rightarrow Volume(V) = 523.59 \;
  $
Therefore, we see that volume of sphere of radius $ 5cm $ is $ 523.59\,\,c{m^3} $
Now, to find the surface area of the sphere. We use a mensuration formula.
Surface Area = $ 4\pi {r^2} $
Substituting value of ‘r’ in above formula we have,
$
\Rightarrow Surface\,\,area\,(A) = 4 \times \dfrac{{22}}{7} \times {(5)^2} \\
\Rightarrow Surface\,\,area\,(A) = 4 \times \dfrac{{22}}{7} \times 5 \times 5 \\
\Rightarrow Surface\,\,area\,(A) = \dfrac{{2200}}{7} \\
\Rightarrow Surface\,\,area\,(A) = 314.15 \\
  $
Therefore, from above we see that the surface area of the sphere is $ 314.15\,c{m^2} $ .
So, the correct answer is “Option D”.

Note: In mensuration problems using values of $ \pi \,\,as\,\,\dfrac{{22}}{7}\,\,or\,\,3.14 $ will cause an approximate change in answer. So students will use the same value of $ \pi $ if mentioned in the problem and if it is not mentioned then students can take the value of $ \pi $ with which calculation of problem seems to be done easily by using it.