College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 3 - Polynomial and Rational Functions - Exercise Set 3.6 - Page 421: 75

Answer

Between 0 and 0.5 seconds

Work Step by Step

Given: The position function s (t) = -16t^2+ v0 t+ s0 Here, v0 = 8 feet per second and s0 = 87 feet Explanation: In order to find the period of time during which the ball’s height exceeds that of the cliff, let us solve the function s (t) = -16t^2+ v0 t+ s0 On substituting the given data, we have s (t) = -16t^2+ 8 t+ 87 Here are the steps required for Solving Polynomial Inequalities: Step 1: One side must be zero and the other side can have only one fraction, so we simplify the fractions if there is more than one fraction. Thus, -16t^2+ 8t > 0 Step 2: Critical or Key Values are first evaluated. In order to this, set the equation equal to zero and then simplified inequality is solved. -16t^2+ 8t = 0 -8 t (2t-1) = 0 -8 t = 0 or 2t-1 = 0 This implies t = 0 or t = 0.5 Step 3: Locate the boundary points on a number line found in Step 2 to divide the number line into intervals. The boundary points are shown as follows: The boundary points divide the number line into three intervals: (-infinity, 0), (0, 0.5), (1.5, infinity) For our purposes the mathematical model is useful only from t = 0 until the diver hits the ground. Let us determine the time when the diver hits the ground. s (t) = 0 -16t^2+ v0 t+ s0 = 0 -16t^2+ 8 t+ 87 = 0 This implies 4t-1 = 9.38 t ≈ 2.6 As t ≥ 0only t = 2.6 fits. Therefore, we use the intervals (0, 0.5), ( 0.5, 2.6). Step 4: Now, one test value within each interval is chosen and f is evaluated at that number. Interval Test value Substitute into f(t) = - 16t^2 + 8 t Conclusion (0, 0.5) 0.25 f(0.25) = - 16(0.25)^2 + 8(0.25) f(t) > 0 = 1, Positive (0, 2.6) 1 f(0.25) = - 16(0.25)^2 + 8(0.25) = - 8, Negative f(t) < 0 Step 5: Write the solution set, selecting the interval or intervals that satisfy the given inequality f (t) > 0 [f(t) = - 16t^2 + 8 t] Based on our work done in Step 4, we see that f (t) > 0 for all x in (0, 0.5). Conclusion: This means that the diver’s height exceeds that of the cliff between 0 and 0.5 second.
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