Question

The ratio of the volume and surface area of a sphere of unit radius:
$ \left( a \right)4:3 \\
  \left( b \right)3:4 \\
  \left( c \right)1:3 \\
  \left( d \right)3:1 $

Answer 1

Hint: In this question, we use the formula of volume and surface area of the sphere. First, find the value of volume and surface area of the sphere then take a ratio. We use volume of sphere $ = \dfrac{{4\pi {r^3}}}{3}$ and also surface area of sphere \[ = 4\pi {r^2}\] where r is the radius of sphere.

Complete step-by-step solution:

Given, radius of sphere r=1
Now use formula of volume of sphere, $V = \dfrac{{4\pi {r^3}}}{3}$
$
   \Rightarrow V = \dfrac{{4\pi \times {{\left( 1 \right)}^3}}}{3} \\
   \Rightarrow V = \dfrac{{4\pi }}{3}.............\left( 1 \right) \\
$
 Now use formula of surface area of sphere, \[S = 4\pi {r^2}\]
$
   \Rightarrow S = 4\pi \times {\left( 1 \right)^2} \\
   \Rightarrow S = 4\pi ................\left( 2 \right) \\
 $
Now, take a ratio of the volume and surface area of a sphere.
Divide (1) equation by (2) equation,
$
   \Rightarrow \dfrac{V}{S} = \dfrac{{\dfrac{{4\pi }}{3}}}{{4\pi }} \\
   \Rightarrow \dfrac{V}{S} = \dfrac{{4\pi }}{{4\pi \times 3}} \\
   \Rightarrow \dfrac{V}{S} = \dfrac{1}{3} \\
$
The ratio of the volume and surface area of a sphere is 1:3.
So, the correct option is (c).

Note: In such types of problems we can use two different ways to solve questions in easy ways. First way we already mentioned above and in a second way, we take a ratio of the volume of a sphere to the surface area of the sphere so we get the general ratio of volume to the surface area of a sphere and then put the value of the radius of the sphere.
$ \Rightarrow \dfrac{V}{S} = \dfrac{{\dfrac{{4\pi {r^3}}}{3}}}{{4\pi {r^2}}} = \dfrac{r}{3}$
So, we can directly use this ratio and get the solution in just seconds.