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# lec12 - Lecture 12 18.01 Fall 2006 Lecture 12 Related Rates...

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Lecture 12 18.01 Fall 2006 Lecture 12: Related Rates Example 1. Police are 30 feet from the side of the road. Their radar sees your car approaching at 80 feet per second when your car is 50 feet away from the radar gun. The speed limit is 65 miles per hour (which translates to 95 feet per second). Are you speeding? First, draw a diagram of the setup (as in Fig. 1): Road Car Police 30 D=50 x Figure 1: Illustration of example 1: triangle with the police, the car, the road, D and x labelled. Next, give the variables names. The important thing to figure out is which variables are changing. dD At D = 50, x = 40. (We know this because it’s a 3-4-5 right triangle.) In addition, = D = dt - 80. D is negative because the car is moving in the - x direction. Don’t plug in the value for D yet! D is changing, and it depends on x . The Pythagorean theorem says 30 2 + x 2 = D 2 Di ff erentiate this equation with respect to time (implicit di ff erentiation: d ( 2 ) 2 DD 30 2 + x = D 2 = 2 xx = 2 DD = x = dt 2 x Now, plug in the instantaneous numerical values: 50 feet x = 40 ( - 80) = - 100 s

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