A projectile is fired straight up from ground level with an initial velocity of $112 ft / s$. Its height, $h$, above the ground after $t$ seconds is given by $h=-16 t^2+112 t$. What is the interval of time during which the projectile's height exceeds 192 feet?
A. $3 < t < 4$
B. $t<3$ or $t>4$
C. $t>4$
D. $3<t<4$
Answer (2)
Set up the inequality 192"> − 16 t 2 + 112 t > 192 .
Rearrange the inequality to t 2 − 7 t + 12 < 0 .
Factor the quadratic to find the roots: ( t − 3 ) ( t − 4 ) = 0 , so t = 3 and t = 4 .
Determine the interval where the inequality holds: 3 < t < 4 .
The interval of time during which the projectile's height exceeds 192 feet is \boxed{3 192 .
Rearranging the Inequality First, let's rearrange the inequality: 192"> − 16 t 2 + 112 t > 192 0"> − 16 t 2 + 112 t − 192 > 0 Now, we can divide the entire inequality by -16. Remember that when we divide by a negative number, we need to reverse the inequality sign: t 2 − 7 t + 12 < 0
Finding the Roots Next, we need to find the roots of the quadratic equation t 2 − 7 t + 12 = 0 . We can factor the quadratic expression: ( t − 3 ) ( t − 4 ) = 0 So, the roots are t = 3 and t = 4 .
Determining the Interval Now we need to determine the interval where t 2 − 7 t + 12 < 0 . Since the parabola opens upwards (the coefficient of t 2 is positive), the quadratic expression is negative between the roots. Therefore, the inequality holds true for 3 < t < 4 .
Final Answer The projectile's height exceeds 192 feet between 3 and 4 seconds. Therefore, the interval of time during which the projectile's height exceeds 192 feet is 3 < t < 4 .
Examples
Understanding projectile motion is crucial in many fields, such as sports and engineering. For example, when designing a catapult or launching a rocket, engineers need to calculate the trajectory and height of the projectile. By solving quadratic equations and inequalities, they can determine the optimal launch angle and velocity to achieve the desired range and height. In sports, athletes and coaches use similar principles to optimize performance in activities like throwing a ball or kicking a football. The concepts of projectile motion help in predicting the path of the object and making necessary adjustments to achieve the best results.
The height of the projectile exceeds 192 feet during the interval 3 < t < 4 . This is derived from solving the quadratic inequality based on the height equation. Therefore, the correct answer is option D.
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