SECTION 10.7 Plane Curves and Parametric Equations 741 Now Work PROBLEM 49 Need to Review? The vertex of a quadratic function y f x ax bx c 2 ( ) = = + + is the point b a f b a 2 , 2 ( ) ( ) − − Refer to Section 3.3, pp. 160–161. (b) To find the length of time that the ball was in the air, solve the equation y t 0. ( ) = t t t t t t 16 75 0 16 75 0 0 sec or 75 16 4.6875 sec 2 ( ) − + = − + = = = = The ball struck the ground after 4.6875 seconds. (c) Notice that the height y of the ball is a quadratic function of t, so the maximum height of the ball can be found by determining the vertex of y t t t 16 75 . 2 ( ) = − + The value of t at the vertex is t 75 32 2.34375 sec = − − = The ball was at its maximum height after 2.34375 seconds. The maximum height of the ball is found by evaluating the function y t( ) at t 2.34375 seconds. = Maximum height 16 2.34375 75 2.34375 87.89 feet 2 ( ) = − ⋅ + ⋅ ≈ (d) Since the ball was in the air for 4.6875 seconds, the horizontal distance that the ball traveled is x 75 3 4.6875 608.92 feet = ⋅ ≈ (e) Enter the equations from part (a) into a graphing utility with T T min 0, max 4.7, = = and Tstep 0.1. = Use ZOOM-SQUARE to avoid any distortion to the angle of elevation. See Figure 70 for the graph using a TI-84 Plus CE. Figure 70 247 2157 220 630 Now Work PROBLEM 53 Exploration Simulate the motion of a ball thrown straight up with an initial speed of 100 feet per second from a height of 5 feet above the ground. Use PARAMETRIC mode on a TI-84 Plus CE with T T T X X Y min 0, max 6.5, step 0.1, min 0, max 5, min 0, = = = = = = and Ymax 180. = What happens to the speed with which the graph is drawn as the ball goes up and then comes back down? How do you interpret this physically? Repeat the experiment using other values for Tstep. How does this affect the experiment? [Hint: In the projectile motion equations, let v h 90 , 100, 5, 0 θ = ° = = and g 32. = Use x 3 = instead of x 0 = to see the vertical motion better.] Result In Figure 71(a), the ball is going up. In Figure 71(b), the ball is near its highest point. Finally, in Figure 71(c), the ball is coming back down. Figure 71 180 0 5 0 (t ø 0.7) (a) 180 0 5 0 (t ø 3) (b) 180 0 5 0 (t ø 4) (c) Notice that as the ball goes up, its speed decreases, until at the highest point it is zero. Then the speed increases as the ball comes back down. A graphing utility can be used to simulate other kinds of motion as well. Let’s rework Example 5 from Section A.8.
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