Question

If the numerical value of surface area of a sphere is equal to its volume then the radius of the sphere is
(a) 1
(b) 3
(c) 5
(d) 4.7

Answer 1

Hint: Here, in this given question, we have to use the formula of the surface area $ 4\pi {{r}^{2}} $ and the volume $ \dfrac{4}{3}\pi {{r}^{3}} $ of a sphere as equal in numerical value so as to get a relation between the square and the cube of the radius and then get our required answer.

Complete step-by-step answer:
In this given question, we are given that the numerical value of the surface area and the volume of a sphere are equal and are asked to find the radius.
We know that the formula for the surface area of a sphere is $ 4\pi {{r}^{2}} $ and the formula for the volume of the sphere is $ \dfrac{4}{3}\pi {{r}^{3}} $ where r is the radius of the sphere.

As, it is given that the numerical value of the surface area and the volume of the sphere is same, we take,
 $ 4\pi {{r}^{2}}=\dfrac{4}{3}\pi {{r}^{3}}..........(1.1) $
Now, simplifying equation 1.1, we get,
 $ \begin{align}
  & 4\pi {{r}^{2}}=\dfrac{4}{3}\pi {{r}^{3}} \\
 & \Rightarrow r=3..........(1.2) \\
\end{align} $
So, from equation 1.2, we get the value of the radius of the concerned sphere as 3.
Therefore, option (b) 3 is the correct option to this given question.

Note: We should note that we have not included any unit for r in the answer. However, as the relation between the surface area and volume can be defined in terms of r, the numerical value of r would be the same in any units provided that the surface area and volume are also in that unit system.