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NCERT Class 12- Mathematics: Chapter β 6 Application of Derivatives Part 9
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 45Β° 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 ΞAOB, 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 .