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
B. $3>t>4$
C. $t<4$
D. $t>4$
Answer (1)
Rewrite the inequality: t 2 − 7 t + 12 < 0 .
Factor the quadratic equation: ( t − 3 ) ( t − 4 ) = 0 .
Find the roots: t = 3 and t = 4 .
Determine the interval where the inequality is satisfied: \boxed{3 192 .
Rewriting the Inequality First, let's rewrite the inequality: 192"> − 16 t 2 + 112 t > 192 . We can rearrange this to get a quadratic inequality: 16t^2 - 112t + 192"> 0 > 16 t 2 − 112 t + 192 . To make it easier to work with, we can divide the entire inequality by 16: t^2 - 7t + 12"> 0 > t 2 − 7 t + 12 , or t 2 − 7 t + 12 < 0 .
Finding the Roots Now, let's find the roots of the quadratic equation t 2 − 7 t + 12 = 0 . We can factor the quadratic as ( t − 3 ) ( t − 4 ) = 0 . This gives us the roots t = 3 and t = 4 .
Determining the Interval Since the quadratic t 2 − 7 t + 12 has a positive leading coefficient (1), the parabola opens upwards. This means that the quadratic is negative between the roots. Therefore, the inequality t 2 − 7 t + 12 < 0 is satisfied when 3 < t < 4 .
Final Answer The interval of time during which the projectile's height exceeds 192 feet is 3 < t < 4 .
Examples
Understanding projectile motion is crucial in fields like sports and military science. For example, when a basketball player shoots a ball, they need to consider the initial velocity and angle to make a basket. Similarly, in military applications, understanding projectile motion helps in accurately targeting artillery shells. The quadratic equation we solved helps determine when the projectile is at a certain height, which is essential for these applications.
