742 CHAPTER 10 Analytic Geometry Simulating Motion Tanya, who is a long-distance runner, runs at an average speed of 8 miles per hour. Two hours after Tanya leaves your house, you leave in your Honda and follow the same route. See Figure 72. If the Honda’s average speed is 40 miles per hour, how long is it before you catch up to Tanya? Use a simulation of the two motions to verify the answer. EXAMPLE 6 Solution Begin with two sets of parametric equations: one to describe Tanya’s motion, the other to describe the motion of the Honda. We choose time t 0 = to be when Tanya leaves the house. If we choose y 2 1 = as Tanya’s path, then we can use y 4 2 = as the parallel path of the Honda. The horizontal distances traversed in time t Distance Rate Time ( ) = × are x t t x t t Tanya: 8 Honda: 40 2 1 2 ( ) ( ) ( ) = = − You catch up to Tanya when x x . 1 2 = t t t t t t 8 40 2 8 40 80 32 80 80 32 2.5 ( ) = − = − − = − = − − = You catch up to Tanya 2.5 hours after Tanya leaves the house. In PARAMETRIC mode with Tstep 0.01, = simultaneously graph ( ) ( ) ( ) ( ) ( ) = = − = = x t t x t t y t y t Tanya: 8 Honda: 40 2 2 4 1 2 1 2 for t 0 3. ≤ ≤ Figure 73 shows the relative positions of Tanya and the Honda for t t t t 0, 2, 2.25, 2.5, = = = = and t 2.75 = on a TI-84 Plus CE. Figure 72 Time t t 5 2 t 5 0 2 hr t 5 2 5 40 t 52.25 0 0 5 40 t 50 0 0 5 40 t 52 0 0 Figure 73 5 40 t 52.5 0 0 5 40 t 52.75 0 0
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