A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate.

Given: a spherical ball salt, it is dissolving such that the rate of decrease of the volume at any instant is proportional to the surface

To prove: the radius is decreasing at a constant rate

Explanation: Let the radius of the spherical ball of the salt a t any time t be ‘r’.

Let the surface area of the spherical ball be S

Then, S = 4πr²……….(i)

Let V be the volume of the spherical ball,

Then, ……..(ii)

Now as per the given criteria,

Here the negative sign indicates the rate of volume is decreasing.

Or we can write this as

Where k is the proportional constant

Substituting the values from equation (i) and (ii), we get

Now taking out the constant term outside on LHS, we get

Applying the derivatives with respect to t, we get

Cancelling the like terms, we get

Hence the radius of the spherical ball is decreasing at a constant rate.

Hence Proved