Question

A ball is thrown directly upward from a height of 7ft with an initial velocity of 24ft/sec. The function s(t)=−16t²+24t+7 gives the height of the ball, in feet, t seconds after it has been thrown. Determine the time at which the ball reaches its maximum height and find the maximum height.

Answer 1

The ball reaches its maximum height after 0.75 seconds, at approximately 25 feet.

Step-by-step explanation:

To find the time at which the ball reaches its maximum height, we must determine the vertex of the parabolic function s(t)=−16t²+24t+7. Since the coefficient of t² is negative, the parabola opens downward and the vertex represents the highest point of the ball's trajectory. The vertex can be found using the formula t = -b/(2a), where a is the coefficient of t² and b is the coefficient of t.

Using the given function, a = -16 and b = 24. Thus the time of maximum height is:
t = -24/(2×(-16)) = -24/(-32) = 0.75 seconds.

Next, we plug this time into the original function to find the maximum height:
s(0.75) = -16(0.75)² + 24(0.75) + 7 ≈ 25 feet.

Therefore, the ball reaches its maximum height after 0.75 seconds, and that height is approximately 25 feet.