Related Rates

Strategy

Assign symbols to all variables
State, in terms of variables, what is given and what is to be determined
Find necessary mathematical relations
Differentiate with respect to the independent variable taking into account the chain rule
Substitute known values and compute the unknown value

Inflating a ball

A spherical ball is being filled with air at a constant rate of $2 {cm}^{3} / sec$ . How fast is the radius increasing the radius is $3 cm$ ?

Approaching train

An observer is standing at a point O, that is 30 meters away from a railroad. A train moving at a speed of 90 meters per second is approaching along the straight track. How fast is the distance between the train and the observer changing when the train is 50 meters from the observer?

Hot air balloon

A hot air balloon raising straight up from a level field is tracked by a range finder 500 meters from the point of lift off. At the moment the range finder's elevation angle is $\frac{π}{2}$ the angle is increasing at a rate of 0.14 radians per minute. How fast is the balloon rising at that moment?