(4)

Step 1 :

Two bikes are start moving from the same place.

Let the starting point is .

After seconds, the first bike is at point .

After seconds, the second bike is at point .

Let the distance traveled by the first bike is .

Let the distance traveled by the second bike is .

The distance between the two bikes is .

Step 2 :

The speed of the first bike is .

The speed of the second bike is .

Time taken by the second rider to reach is .

The first bike starts early than second bike, so the time taken by first bike to reach is .

Find distance traveled by each bikes.

.

Distance traveled by first bike .

Distance traveled by second bike .

Find distance between two riders .

Solve by using Pythagorean theorem to above figure.

The distance between two riders is .

Step 3 :

Find rate of distance between two bike riders.

To find rate of change of distance between two bike riders differentiate with respect to .

Substitute

, ,  , and in .

Rate of distance between two riders is .

Solution :

Rate of distance between two riders is .

(3)

Step 1 :

Let the rocket is fired from the point .

A person is standing at point .

After seconds, rocket is at point .

Let the distance traveled by the rocket is .

The distance between rocket fired point to person is .

Distance between the person and rocket after seconds .

Step 2 :

The speed rocket is .

Find distance traveled by rocket after .

.

Substitute , in the distance formula.

.

Distance traveled by rocket is .

Distance between rocket fired to person is .

Apply Pythagorean theorem to the figure and Solve .

.

Distance between rocket fired after to person is .

Step 3 :

Find rate of distance between rocket fired after to person.

To find rate of distance between two riders.

Differentiate on each side with respect to .

Since person is standing so .

Substitute , , and in .

Rate of distance between person and rocket is .

Solution :

Rate of distance between person and rocket is .