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

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4 : 3

3 : 4

4 : 3 π

3 : 4 π

answer is A.

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Detailed Solution

Given that the radius and height of the cylinder are the same. That's why we can write

.
It is also given that the radius of the sphere is equal to the height of the cylinder. So we can write

.
Now we need to differentiate the volume of the sphere as a function of time to get the rate of increase of the sphere's volume.
Therefore, the rate of increase in the volume of the sphere.

= d d t 4 π R 3 3 = d d t 4 π h 3 3 = 4 π h 2 3 d h d x

Now we need to differentiate the volume of the cylinder with respect to time to get the rate of increase of the volume of the cylinder.
The rate of increase in cylinder volume.

= d d t π r 2 h = d d t π h 2 h = π h 2 d h d x

Now we need to find the ratio of the rate of increase in the volume of the sphere to the rate of increase in the volume of the cylinder.
Dividing by the increase in volumes, we get
Ratio