Question

The curved surface area of a hemisphere is $905\dfrac{1}{7}c{{m}^{2}}$, what is the volume of the hemisphere?
A.$3580.57c{{m}^{3}}$
B.$8220.57c{{m}^{3}}$
C.$8830.57c{{m}^{3}}$
D.$3620.58c{{m}^{3}}$

Answer 1

Hint: From the given surface area find out the radius of the hemisphere. The formula of the surface area of a hemisphere is $2\pi {{r}^{2}}$ square unit, where r is the radius. After this, find out the volume of the hemisphere and the formula is $\dfrac{2}{3}\pi {{r}^{3}}$ cubic unit.

Complete step-by-step answer:

A sphere is defined as a set of points in three-dimension and all the points lying on the surface are equidistant from the center. When a plane cuts across the sphere at the center, it forms a hemisphere. We can say, a hemisphere is exactly half of a sphere.
Here it is given that the curved surface area of a hemisphere is $905\dfrac{1}{7}c{{m}^{2}}$.
Let the radius of the sphere is r cm.
 We know that the surface area of a sphere is $4\pi {{r}^{2}}$ square unit.
Since the hemisphere is half of a sphere, the surface area of a hemisphere will be $2\pi {{r}^{2}}$ square unit.
Now we can form an equation to find out the radius of the hemisphere. Therefore,
$2\pi {{r}^{2}}=905\dfrac{1}{7}$
$\Rightarrow 2\pi {{r}^{2}}=\dfrac{\left( 905\times 7 \right)+1}{7}$
$\Rightarrow 2\pi {{r}^{2}}=\dfrac{6335+1}{7}$
$\Rightarrow 2\pi {{r}^{2}}=\dfrac{6336}{7}$
Now, divide the both sides of the equation by $2\pi $.
$\Rightarrow {{r}^{2}}=\dfrac{6336}{7\times 2\times \pi }$
Put $\pi =\dfrac{22}{7}$, in the above expression.
$\Rightarrow {{r}^{2}}=\dfrac{6336\times 7}{7\times 2\times 22}$
Now we can cancel out the common factor from the numerator and the denominator. We will get,
$\Rightarrow {{r}^{2}}=144$
$\Rightarrow r={{\left( {{12}^{2}} \right)}^{\dfrac{1}{2}}}$
$\Rightarrow r=12$
Therefore, the radius of the hemisphere is 12 cm.
Now we know that the formula of the volume of a hemisphere is $\dfrac{2}{3}\pi {{r}^{3}}$ cubic unit.
We know the radius. Therefore the volume will be,
$\dfrac{2}{3}\pi {{r}^{3}}=\dfrac{2}{3}\times \dfrac{22}{7}\times {{12}^{3}}=3620.57c{{m}^{3}}$
Hence option (d) is correct.

Note: Here the given figure is a hemisphere. Hemisphere is exactly half of a sphere. Don’t forget to divide the sphere formulas by 2.