Question

The total surface area of a sphere is $452\dfrac{4}{7}c{{m}^{2}}$. Find the volume of the sphere.

Answer 1

Hint: For solving this problem, first we calculate the radius of the sphere by using the formula for total surface area as the value of total surface area of a sphere is given in the problem statement. Once we obtain the radius of the sphere, we can easily calculate the volume by using the standard formula for volume of sphere.

Complete step-by-step solution -
According to the problem statement, we are given a sphere having a total surface area $452\dfrac{4}{7}c{{m}^{2}}$. The total surface area of a sphere can be expressed as: $4\pi {{r}^{2}}$, where r is the radius of the sphere.
On equating the above values, we get radius as:
\[\begin{align}
  & 4\pi {{r}^{2}}=452\dfrac{4}{7} \\
 \Rightarrow & 4\times \dfrac{22}{7}\times {{r}^{2}}=\dfrac{3168}{7} \\
 \Rightarrow & {{r}^{2}}=\dfrac{3168}{7}\times \dfrac{7}{22\times 4} \\
\Rightarrow & {{r}^{2}}=36 \\
\Rightarrow & r=\sqrt{36} \\
\Rightarrow & r=6cm \\
\end{align}\]
Therefore, the radius of a sphere is 6 cm.

We are required to calculate the volume of the sphere. As we know that the volume of the square can be expressed as: $V=\dfrac{4}{3}\pi {{r}^{3}}$
On putting the value of radius as 6 cm, we get
$\begin{align}
  & V=\dfrac{4}{3}\pi {{\left( 6 \right)}^{3}} \\
\Rightarrow & V=\dfrac{4}{3}\times \dfrac{22}{7}\times 6\times 6\times 6 \\
\Rightarrow & V=\dfrac{6336}{7} \\
\Rightarrow & V=905.143c{{m}^{3}} \\
\end{align}$
Therefore, the volume of the sphere is 905.143 ${cm}^{3}$.

Note: Students must remember the formula of total surface area and volume of a sphere for solving this problem. Another important thing in this question is the unit system which is in centimetre. So, the final answer should be left in terms of centimetres.