Math 151 Section 3.10
Related Rates
For certain relation in several variables which all vary over time, find the rate of change of one of the
variables with respect to time given the rates of change of the others and enough other information all
at a specific instant.
Step 1. Set up the relation between the variables. If needed sketch a diagram and identify the variables.
Step 2. Differentiate with respect to time.
Step 3. Substitute the given information at the specific instant. Solve for the desired quantity.
Example1: A right circular cone has base radius, r, and constant height 3m. The radius is increasing at the
rate of 0.2m/s. Find the rate of change with respect to time of the volume of the cone at the instant that
the radius is 1.5 m.
Example 2: For the cone above, suppose the height is also changing. If the height is decreasing by 0.3m/s
and the radius is increasing by 0.2m/s, find the rate of change of the volume at the instant that the
radius is 1.5 m and the height is 3 m.
Example 3: A tank in the shape of a right circular cone, inverted, has a height of 16ft and a base radius
4ft. Water flows into the tank at 2 cubic ft /min. How fast is the water level rising at the instant that the
water is 5 feet deep?
Example 4: A ladder 25 ft long is leaning against a vertical wall. The foot of the ladder is sliding away
from the wall at 3 ft/s. a) How fast is the top of the ladder sliding down the wall at the instant that the
foot is 15 feet from the wall? b) How fast is the angle between the ladder and the horizontal line from
the foot of the ladder to the wall changing at this same instant?
Example 5: A helicopter is flying away from the camera at a speed of 90 ft/min. at a constant height of
400 feet. A camera on the ground is trained on the helicopter. Let θ be the angle between the camera
eye and the horizontal. Find the rate of change of θ with respect to time in minutes at the instant that
the distance between the helicopter and the camera is 500 feet.