AP Calculus
2.6 Related Rates
Example #1
A particle is moving along the parabola y = 2x2 – 6x. Given that
dx
dy
 2 , find
dt
dt
when x = 3.
Steps for solving related-rate problems
1. Identify given quantities
2. Identify quantities to be determined
3. Make a sketch and label
4. Write an equation involving variables whose rate of changes are being used
5. Differentiate both sides of the equation implicitly with respect to time
6. Substitute all known values into previous equation and solve for unknown rate of change
Example #2
Air is being pumped into a spherical balloon so that its volume increases at a rate of
100 cm3 /s . How fast is the radius of the balloon increasing when the diameter is 50
cm?
Example #3
In a conical tank, the height of the fluid inside is changing at a rate of -0.2 ft/min and
the radius is changing at a rate of -0.1 ft/min. What is the rate of change in the
volume when the radius is r = 1 and the height is h = 2 ft?
Example #4
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides
away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down
the wall when the bottom of the ladder is 6 ft from the wall?
Example #5
A water tank has the shape of an inverted circular cone with the base radius 2 m and
height of 4 m. If water is being pumped into the tank at a rate of 2 m3 /min , find the
rate at which the water level is rising when the water is 3 m deep.
Example #6
A plane flying horizontally at an altitude of 3 mi and a speed of 500 mi/h passes
directly over a radar station. Find the rate at which the distance from the plane to the
station is increasing when it is 4 mi away from the station.
Example #7
A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the
ground 20 ft from the path and is kept focused on the man. At what rate is the
searchlight rotation when the man is 15 ft from the point on the path closest to the
searchlight?
Example #8
Car A is traveling west at 50 mi/h and car B is traveling north at 60 mi/h. Both are
headed for the intersection of the two roads. At what rate are the cars approaching
each other when car A is 0.3 mi and car B is 0.4 mi from the intersection?
Example #9
A baseball diamond is a square with sides of 90 ft. A batter hits the ball and runs
towards first base with a speed of 24 ft/s. Find the following.
a) The rate at which the runner’s distance from 2nd base is decreasing when
he is halfway to 1st base.
b) At what rate is his distance from 3rd base increasing at the same moment?
Example #10 At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and
ship B is sailing north at 25 km/h. How fast is the distance between the ships
changing at 4 pm?