4 Solve Applied Problems Using Given Quadratic Equations Let’s see how to use a quadratic equation to model a real-life situation. EXAMPLE 5 In-Class Example 5 Use Example 5. An object is launched from a platform with an initial velocity of 32 ft/sec. The height h (in feet) of the object t seconds after it is released is given by the quadratic equation h 16t2 32t 20 a) What is the initial height of the ball? b) How long does it take the ball to reach a height of 32 feet? c) How long does it take for the ball to hit the ground? Solution a) Since t represents the number of seconds after the ball is thrown, t 0 at the time of release. Let t 0, and solve for h. h 16(0)2 32(0) 20 Substitute 0 for t. h 0 0 20 h 20 The initial height of the ball is 20 ft. b) We must fi nd the time it takes for the ball to reach a height of 32 feet. Find t when h 32. h 16t2 32t 20 32 16t2 32t 20 Substitute 32 for h. 0 16t2 32t 12 Write in standard form. 0 4t2 8t 3 Divide by 4. 0 (2t 1)(2t 3) Factor. b R 2t 1 0 or 2t 3 0 Set each factor equal to 0. 2t 1 2t 3 t 1 2 or t 3 2 Solve. How can two answers be possible? After 1 2 sec, the ball is 32 feet above the ground on its way up, and after 3 2 sec, the ball is 32 feet above the ground on its way down. The ball reaches a height of 32 feet after 1 2 sec and after 3 2 sec. Draw a picture to visualize what this situation might look like. www.mhhe.com/messersmith SECTION 7.6 Applications of Quadratic Equations 437
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