Question
Suppose h(t)= -0.2t² + 2t models the height, in feet, of a ball that is kicked into the air where t is given as time, in seconds.
After how many seconds does the ball reach its maximum height? ___ seconds
What is the maximum height of the ball ___ feet ?
feet After how many seconds does the ball reach the ground ___ seconds ?
After how many seconds does the ball reach its maximum height? ___ seconds
What is the maximum height of the ball ___ feet ?
feet After how many seconds does the ball reach the ground ___ seconds ?
Answers
GPT 3.5
To find the maximum height of the ball, we need to first determine the time it takes for the ball to reach its maximum height. This can be done by finding the vertex of the parabolic function h(t) = -0.2t² + 2t.
The t-coordinate of the vertex of a parabola with equation y = ax² + bx + c is given by t = -b/2a.
In this case, a = -0.2 and b = 2. Plugging these values into the formula, we get:
t = -2 / (2 * -0.2)
t = -2 / -0.4
t = 5
Therefore, the ball reaches its maximum height after 5 seconds.
To find the maximum height, we need to substitute t = 5 into the function h(t):
h(5) = -0.2(5)² + 2(5)
h(5) = -0.2(25) + 10
h(5) = -5 + 10
h(5) = 5 feet
Therefore, the maximum height of the ball is 5 feet.
To find when the ball reaches the ground, we need to set h(t) equal to 0 and solve for t:
0 = -0.2t² + 2t
0 = -0.2t(t - 10)
This implies that either t = 0 or t = 10. Since the ball is initially kicked into the air, the time at t = 0 is not relevant. Therefore, the ball reaches the ground after 10 seconds.
The t-coordinate of the vertex of a parabola with equation y = ax² + bx + c is given by t = -b/2a.
In this case, a = -0.2 and b = 2. Plugging these values into the formula, we get:
t = -2 / (2 * -0.2)
t = -2 / -0.4
t = 5
Therefore, the ball reaches its maximum height after 5 seconds.
To find the maximum height, we need to substitute t = 5 into the function h(t):
h(5) = -0.2(5)² + 2(5)
h(5) = -0.2(25) + 10
h(5) = -5 + 10
h(5) = 5 feet
Therefore, the maximum height of the ball is 5 feet.
To find when the ball reaches the ground, we need to set h(t) equal to 0 and solve for t:
0 = -0.2t² + 2t
0 = -0.2t(t - 10)
This implies that either t = 0 or t = 10. Since the ball is initially kicked into the air, the time at t = 0 is not relevant. Therefore, the ball reaches the ground after 10 seconds.