Question

The radius and height of a cylinder are equal. If the radius of the sphere is equal to the height the cylinder, then the ratio of the rates of increase of the volume of the sphere and the volume of the cylinder is
A) \[4:3\]
B) \[3:4\]
C) \[4:3{\pi}\]
D) \[3:4{\pi}\]

Answer 1

Hint:
Here we will differentiate the volumes of both cylinder and sphere with respect to time to get the rate of the increase of the volumes. Then, we will divide the obtained rate of the increase of the volume of the sphere with the rate of the increase of the volume of the cylinder. It will give us the required ratio.
Formula Used: We will use the formula of:
1) Volume of sphere, \[\dfrac{{4\pi {R^3}}}{3}\] , where, \[R\] is the radius of the sphere.
2) Volume of the cylinder, \[\pi {r^2}h\] , where, r is the radius of the cylinder, h is the height of the cylinder

Complete step by step solution:
It is given that the radius and height of a cylinder are equal. Therefore, we can write \[r = h\]. Also it is given that the radius of the sphere is equal to the height of the cylinder. So, we can write \[R = h\].
Now we have to differentiate the volume of the sphere with respect to time to get the rate of the increase of the volume of the sphere.
Therefore, rate of the increase of the volume of sphere\[ = \dfrac{d}{{dt}}\left( {\dfrac{{4\pi {R^3}}}{3}} \right) = \dfrac{d}{{dt}}\left( {\dfrac{{4\pi {h^3}}}{3}} \right) = \dfrac{{4\pi {h^2}}}{3}\dfrac{{dh}}{{dx}}\]
Now we have to differentiate the volume of the cylinder with respect to time to get the rate of the increase of the volume of the cylinder.
Rate of the increase of the volume of cylinder\[ = \dfrac{d}{{dt}}\left( {\pi {r^2}h} \right) = \dfrac{d}{{dt}}\left( {\pi {h^2}h} \right) = \pi {h^2}\dfrac{{dh}}{{dx}}\]
Now, we have to find out the ratio of the rate of the increase of the volume of the sphere to the rate of the increase of the volume of the cylinder.
Dividing the increase of volumes, we get
Ratio \[ = \dfrac{{\dfrac{{4\pi {h^2}}}{3}\dfrac{{dh}}{{dx}}}}{{\pi {h^2}\dfrac{{dh}}{{dx}}}} = \dfrac{4}{3}\]
Hence, \[\dfrac{4}{3}\] is the ratio of the rates of increase of the volume of the sphere and the volume of the cylinder

So, option A is correct.

Note:
Whenever the word rate comes that means the value of the variable varies with respect to time i.e. it changes with the change of time. So to find the rate of change we have to differentiate the equation with respect to time.