Sec 2.6 Related Rates In related rates problems, one tries to find the rate at which some quantity is changing by relating it to other quantities whose rates of change are known. Read intro on p.149 Strategy for Solving Related Rates Problems 1. Understand the problem---identify the variable whose rate of change is known and the variable whose rate you need to find. 2. Develop a mathematical model of the problem. Note: This may involve drawing a picture. 3. Write an equation relating the variable whose rate of change is to be found with the variable(s) whose rate(s) of change(s) are known. 4. Differentiate both sides of the equation implicitly with respect to time, t, and solve for the derivative that will give the unknown rate of change. 5. Evaluate this derivative by substituting values for any quantities that depend on time. Note: only do this AFTER you differentiate. 6. Interpret your solution.

Let t = number of seconds elapsed from the time of the spill r = radius of the spill in feet after t seconds A = area of the spill in square feet after t seconds Ex1. Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2 ft/sec. How fast is the area of the spill increasing when the radius of the spill is 60 ft? Since the oil is spreading in a circular pattern we will use. Because A and r are functions of t, we can differentiate both sides implicitly with respect to time. Why is the unit square feet/sec?

Ex2. A baseball diamond is a square whose sides are 90 ft. long. Suppose that a player running from 2nd to 3 rd has a speed of 30 ft./sec. at the instant when he is 20 ft. from 3rd base. At what rate is the player's distance from home plate changing at that instant? Let t = number of seconds after the player leaves 2nd base x = distance in feet FROM 3rd base y = distance in feet FROM home plate So what do you call the RATE at which the distance from 3 rd base changes? What do you call the RATE at which the distance from home plate changes? What rate are you trying to find? What equation can we use to relate x and y? differentiate w/respect to t. So what values do we have to substitute? x=20 ft, dx/dt = 30 ft/sec. We also need a value for y! Why Neg?

Ex3. A hot-air balloon rising straight up from a level field is tracked by a range finder, 500 ft. 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 minute. How fast is the balloon rising at that moment? Let h be the height of the balloon and Θ be the elevation angle. substitute: Θ h 500 ft EQ? Differentiate w/respect to t: