Warm Up Page 251 Quick Review 1-6 Reference page for Surface Area & Volume formulas

4.6 Related Rates What you’ll learn about… Related Rate Equations Solution Strategies Simulating Related Motion Why? Related rate problems are at the heart of Newtonian mechanics; it was essentially to solve such problems that calculus was invented.

What are Related Rate Equations? Any equation involving two or more variables that are differentiable functions of time “t” can be used to find an equation that relates their corresponding rates. We use implicit differentiation to differentiate several variables with respect to time.

Example 1: Finding Related Rate Equations a)Assume that the radius r of a sphere is a differentiable function of t and let V be the volume of the sphere. Find an equation that relates dV/dt and dr/dt. V = dV/dt = b)Assume that the radius r and height h of a cone are differentiable functions of t and let V be the volume of the cone. Find an equation that relates dV/dt, dr/dt, and dh/dt. V = dV/dt = You try: Given x 2 + y 2 = z 2. Find an equation that relates dx/dt, dy/dt, and dz/dt.

Strategy for Solving Related Rate Equations 1.Define all variables. 2.Draw a picture and write an equation relating the variables. Be sure to distinguish constant quantities from variables that change over time! 3.Write an equation relating the variables. 4.Differentiate the equation with respect to time. 5.Substitute values for any quantities that depend on time. Then simplify / solve the equation. 6.Interpret your answer in a sentence. Don’t forget the units!

What are related rate equations and what can they tell us? You Tube Just Math Tutoring Related Rate Equations Examples 1 & 2

Example 2: A hot-air balloon rising straight up from a level field is tracked by a range finder 500’ from the lift off point. At the moment the range finder’s elevation angle is π/4, the angle is increasing at the rate of 0.14 radians per minutes. How fast is the balloon rising at that moment? 1.Identify variables 2.Draw picture – label 3.Find formula 4.Differentiate implicitly 5.Substitute explicit values into differentiated formula 6.Interpret solution in a sentence

Example 3 A police cruiser, approaching a right-angles intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car? 1.Identify variables 2.Draw picture – label 3.Find formula 4.Differentiate implicitly 5.Substitute explicit values into differentiated formula 6.Interpret solution in a sentence

You try Exercise 13 An Airplane is flying at an altitude of 7 mi and passes directly over a radar antenna as shown in the figure on p 251. When the plane is 10 mi from the antenna (s = 10), the radar detects that the distance s is changing at the rate of 300 mph. What is the speed of the airplane at that moment? 1 & 2 Identify variables / Draw picture – label 3. Find formula 4. Differentiate implicitly 5. Substitute explicit values into differentiated formula 6. Interpret solution in a sentence

Homework Page 251 Exercises 1-3, 7, 8, 9, 11, 13, 15 Follow steps & Answer questions with units!!!

Review: What are related rate equations and how do we solve them? You tube: justmathtutoring Related rate equation example

Example 4 Water runs into a conical tank at the rate of 9 ft 3 /min. The tank stands point down and has a height of 10’ and a base radius of 5’. How fast is the water level rising when the water is 6’ deep? 1.Identify variables 2.Draw picture – label 3.Find formula 4.Differentiate implicitly 5.Substitute explicit values into differentiated formula 6.Interpret solution in a sentence (note 2 nd solution strategy on p 249)

What now? Free Response Question – Graded Work with a partner, each student turn in their own paper. Homework Page 253 Exercises 21, 24, 29, 34, (discuss picture for #29) + Study for quiz 4.4 thru 4.6