1. Related Rates
ex: Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm3/s. How fast is the radius of the balloon increasing when the diameter is 50cm?
Identify known values:
V rate of increase = 100 (cm3/s)
Assign this to be dV/dt
Identify unknown values:
Rate of increase of the diameter = 50.
Assign this to be dr/dt
Find formula to relate the known & unknown quantities:
V = 4/3r3
Differentiate both sides of the formula w/respect to t…

2. V = 4/3r3 =100 snapshot @ diameter = 50r = 25
unknown we are solving for
diameter = 50r = 25

3. ex: A 15 foot ladder is resting against the wall
ex: A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of ¼ ft/sec.  How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?
snapshot
t = 12
Diagram
1) Identify known values x y z
= 15
= 0
= unknown
= 10 (starting distance)
= -.25

4. Values after snapshot

5. 3) Find a formula to relate the known & unknown quantities
4) Differentiate both sides of the formula w/respect to t:
5) Fill in knowns and snapshot – isolate unknown:

6. A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. 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 3m deep.
Known
Unknown
Diagram:
Snapshot
Formula
Derivative

7. Replacing variables
What is the relationship between r and h?
r = 2
h = 4
Try the derivative again…

8. Fill-in knowns and snapshot…
Solve for unknown…

9. A man walks along a straight path at a speed of 4 ft/s
A man walks along a straight path at a speed of 4 ft/s. A rotating searchlight located 20 ft away at point perpendicular to the path follows the man. At what rate is the light rotating when the man is 15 feet from the point on the path that is perpendicular to the light? 20 x
Known
Unknown
Snapshot
Formula

10. Not needed at the moment…
Now do derivative…
Fill-in knowns and snapshot…
Known
Snapshot
Not needed at the moment…
Solve for unknown…
Recall that

11. 20 15
(3)
Need cos q to solve this…
Use snapshot…
Snapshot
(4)
= 5

12. Steps for solving related rates problems
Identify knowns, unknowns and snapshot.
Find a formula that relates knowns and unknowns.
Take the derivative of the formula on both sides w/respect to t.
Plug known values and snapshot into derivative.
Solve for unknown.