# NCERT Class 12-Mathematics: Chapter –6 Application of Derivatives Part 11

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Question 9:

A swimming pool is to be drained for cleaning. If represents the number of litres of water in the pool seconds after the pool has been plugged off to drain and . How fast is the water running out at the end of seconds? What is the average rate at which the water flows out during the first seconds?

Let L represents the number of litres of litres of water in the pool seconds after the pool been plugged off to drain, then

Rate at which the water is running out

Since,

Question 10:

The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.

Given: a volume of cube increasing at a constant rate

To prove: the increase in its surface area varies inversely as the length of the side

Explanation: Let the length of the side of the cube be ‘a’.

Let V be the volume of the cube,

Then

As per the given criteria the volume is increasing at a uniform rate, then

Now substituting the value from equation (i) in above equation, we get

Now differentiating with respect to t we get

Now let S be the surface area of the cube, then

Now differentiating surface area with respect to , we get

Applying the derivatives, we get

Now substituting value from equation (ii) in the above equation we get

Cancelling the like terms we get

Converting this to proportionality, we get

Hence the surface area of the cube with given condition varies inversely as the length of the side of the cube.

Hence Proved

Question 11:

and are the sides of two squares such that . Find the rate of change of the area of Second Square with respect to the area of First Square.