Calculus
Ch 4.6 Related Rates Notes
Name______________________________
Date _________________________
Rate of Change of Q =
dQ
dt
Steps for solving related rates problems:
1. Understand the situation. Draw and label a diagram. Distinguish between constant values and
those that change. Only label those values that are constant.
2. Identify changing quantities and restate the problem. Then exchange the words with dQ/dt to
represent the changing rates where Q is the variable that is changing. Use the word model:
Find the rate of change of ______ given that the rate of change of _______ is ____
at the instant _____________ is ______.
3. Write an equation that relates the changing quantities. Usually this is some geometric formula
like area, volume, the Pythagorean formula, etc.
4. Implicitly derive the formula by taking d/dt of each term.
5. Substitute the given values into the derived formula. Only do this after you have derived the
formula!
6. Interpret the results.
Ex. 1) The area of a circular doggie puddle is growing at a rate of 12 cm 2 / s . How fast is the radius
growing at the instant when the radius is 10 cm?
Ex. 2) Joey is at the top of a 10 ft. ladder which leaning against a wall. The ladder starts to side
down the wall at a rate of 4 ft/min. Lou is standing on the ground 6 ft. away from the wall. How fast
is the base of the ladder moving when it his Lou?
Ex. 3) Two cars starting moving from the same point. One travels South at 25 mph and the other
travels West at 15 mph. At what rate ist he distance between the cars increasing 2 hours later?
Ex. 4) A splotlight on the ground shines on a building 12m away. A man 2m tall walks away from the
spotlight towards the building at a speed of 1.6 m/sec. How fast is the length of his shadow
decreasing when he is 4 m from the building?
Ex. 5) A water tank has the shape of an inverted circular cone with base radius of 2m and height of
4m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water is
1
rising when the water is 3m deep. Volume of cone formula: V   r 2 h
3
2
4
m