4.6 – Related Rates “Trees not trimmed don't make good timber; children not educated don't make useful people.” Unknown http://quotes.dictionary.com Warm – Up: Find an equation of the tangent line to the graph at the given point. #26 from pg.297

4.6 Day 1 Objectives Discuss what related rates are. Find the derivatives using implicit differentiation, like 4.5. Apply our skills to solve real-life type of problems.

What are related rates? We have been finding rates of change with respect to x, and will now find rates of change with respect to t (time). Article to view at home: “The Lengthening Shadow: The story of related rates.” by Bill Austin, Don Barry and David Berman. www.matharticles.com

Example 1 Assume x and y are differentiable functions of t and find the required values of dy/dt and dx/dt. Equation Find Given When x = 3

Equation Find Given when x = 1

Example 2 pg.301 A peddle is dropped in a pond, causing ripples in the form of concentric circles, as shown. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?

General Guidelines, pg.301 Identify all given quantities. Write an equation involving the variables & situation. (rate of changes involved) Using the chain rule, implicitly differentiate both sides with respect to t. Substitute with given information, then solve for the required rate of change. Review pgs.301 & 302 “boxes of information” outside of class.

Example 3 A 25 foot ladder is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. A) How fast is the top of the ladder moving down the wall when its base is 7 feet from the wall?

B) Consider the triangle formed by the side of the house, ladder and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall.

Last Example #32 from HW Air Traffic Control An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. When the plane is 10 miles away (s = 10), the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the plane?

4.6 Day 2 - Related Rates “Success is the maximum utilization of the ability that you have.” Zig Ziglar To do today: More related rates examples. Finish AP Questions if time permits.

Example 1 Volume A spherical balloon is inflated with gas at the rate of 800 cm3/min. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 cm and (b) 60 cm?

Example 2 A baseball diamond has the shape of a square with sides 90 feet long. A player running from 2nd to 3rd base at a speed of 28 ft. per second is 30 feet from third base. At what rate is the player’s distance s from home plate changing?

Homework & reminders HW: 4.6 Day 2 Assignment (skip #25 & 35) Bring your Fast Track books for tomorrow and Wednesday. Friday is the Ch.4 Test