2. 4.1 - Related Rates ex: Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm 3 /s. How fast is the radius of the balloon increasing when the diameter is 50cm? Step 1: Collect problem info… KnownUnknownSnapshot D = 50 Step 2: Write formula that relates known and unknown…

3. Step 3: Take derivative of formula on both sides w/respect to ‘t’ … Step 4: Plug-in knowns and snapshot… Step 5: Solve for unknown… Diameter = 50 implicit derivative (placeholder)

4. 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? x y z = 10 (starting distance) Step 1: Diagram and Label (Cartesian) Step 2: Collect Problem Info Snapshot: t = 12 sec

5. Values after snapshot

6. 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:

7. IC 4.1A – pg 272 #’s 1-7 all

8. Derivative 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 m 3 /min, find the rate at which the water level is rising when the water is 3m deep. KnownUnknown SnapshotFormula Diagram:

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

10. Fill-in knowns and snapshot… Known Snapshot Solve for unknown…

11. 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 KnownUnknown SnapshotFormula Probably better to NOT have a quotient in our formula… x = 15

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

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

14. IC 4.1B – pg 272 #’s 14, 17,18,21,22

15. Calc 4.1 - Steps for solving related rates problems 1.Identify knowns, unknowns and snapshot. 2.Find a formula that relates knowns and unknowns. 3.Take the derivative of the formula on both sides w/respect to t. 4.Plug known values and snapshot into derivative. 5.Solve for unknown.