(d) How high will the rock be 1.5 seconds after it is launched? (e) What is the maximum height attained by the rock? After how many seconds will this happen? Deter- mine the answer analytically and graphically. (f) After how many seconds will the rock hit the ground? Determine the answer graphically. 66. Height of a Toy Rocket A toy rocket is launched from the top of a building 50 feet tall at an initial velocity of 200 feet per second. Lett represent the amount of time p elapsed after the launch. (a) Express the height s as a function of the time t. (b) Determine both analytically and graphically the time at which the rocket reaches its highest point. How high will it be at that time? (c) For what time interval will the rocket be more than 300 feet above ground level? Determine the answer graphically, and give times to the nearest tenth of a second. (d) After how many seconds will the rocket hit the ground? Determine the answer graphically. bongo 67. Height of a Projected Ball A ball is launched upwardan from ground level with an initial velocity of 150 feet per ner second. (a) Determine graphically whether the ball will reach a height of 355 feet. If it will, determine the time(s) when this happens. If it will not, explain why, using niet a graphical interpretation. OWT) 3.3 Quadratic Equations and Inequalities 187 (b) Repeat part (a) for a ball launched from a height of 30 feet with an initial velocity of 250 feet per second. 68. Height of a Projected Ball on the Moon An astronaut on the moon throws a baseball upward. The astronaut is 6 feet, 6 inches tall and the initial velocity of the ball is 30 feet per second. The height of the ball is approximated by the function s(t) = -2.712 + 30r + 6.5, where is the number of seconds after the ball was thrown. (a) After how many seconds is the ball 12 feet above the moon's surface? (b) How many seconds after it is thrown will the ball return to the surface? (c) The ball will never reach a height of 100 feet. How can this be determined analytically? Concept Check Sketch a graph of a quadratic function that satisfies each set of given conditions. Use symmetry to label another point on your graph. 69. Vertex (-2,-3); through (1,4) 70. Vertex (5, 6); through (1,-6) 71. Maximum value of 1 at x = 3; y-intercept is (0,-4) 72. Minimum value of -4 at x = -3; y-intercept is (0, 3) 0 = (8 noitulo2 oilylenA 3.3 Quadratic Equations and Inequalities Zero-Product Property Square Root Property and Completing the Square Quadratic Formula and the Discriminant Solving Quadratic Equations Solving Quadratic Inequalities Formulas Involving Quadratics