# NCERT Class 12-Mathematics: Chapter –6 Application of Derivatives Part 9

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**Question 4:**

Two men A and B start with velocities *v* at the same time from the junction of two roads inclined at to each other. If they travel by different roads, find the rate at which they are being separated.

**Answer:**

Given: two men A and B start with velocities v at the same time from the junction of the two roads inclined at 45° to each other

To find the rate at which they are being separated

Explanation:

Let A and B move a distance of x on different roads as shown above, there distance at any time t will be same as they have same velocity.

Hence

Now consider ΔAOB, applying the cosine rule, we get

Now multiplying and dividing by , we get

Now applying the derivative with respect to , we get

Taking out the constant terms, we get

Substituting the value from equation (i), we get

Hence this is the rate at which the two roads are being separated

**Question 5:**

Find an angle which increases twice as fast as its sine.

**Answer:**

Given: a condition

To find the angle such that it increases twice as fast as its sine.

Explanation: Let

On differentiating with respect to , we get

Applying the derivative, we get

But it is given that

Substituting this value in equation (i), we get

Now cancelling the like terms, we get

But given this is possible only when

Hence the angle is .