The height h (in feet) of a fly ball in a baseball game can be modeled by h=-16^2+28+8, where t is time.
A. Do both t-intercepts of the graph of the function have meaning in this situation? Explain?
B. No one caught the fly ball. After how many seconds did the ball hit the ground?
There's a typo. It should read
h = -16t^2 + 28t + 8
A. No. The ball starts at a height of 8 ft, so the negative x-intercept occurs before it was hit.
B. Just solve for t when h = 0
A. In this situation, the t-intercepts of the graph do not have a meaningful interpretation. This is because the equation h = -16t^2 + 28t + 8 is a quadratic equation that models the height of the fly ball as a function of time. The t-intercepts correspond to the points where the height is equal to zero, which means the ball would hit the ground. However, in this case, the t-intercepts of the graph do not have real solutions, indicating that the ball does not hit the ground during the time interval being considered. Therefore, the t-intercepts of this graph do not have any meaningful interpretation in terms of the ball hitting the ground.
B. To determine when the ball hits the ground, we need to find the value of t for which h = 0. So, we can set the equation -16t^2 + 28t + 8 = 0 and solve for t.
To solve the quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -16, b = 28, and c = 8. We can substitute these values into the quadratic formula:
t = (-28 ± √(28^2 - 4(-16)(8))) / (2(-16))
Simplifying further:
t = (-28 ± √(784 + 512)) / (-32)
t = (-28 ± √1296) / (-32)
t = (-28 ± 36) / (-32)
Considering both possibilities:
t1 = (-28 + 36) / (-32) = 8 / (-32) = -0.25
t2 = (-28 - 36) / (-32) = -64 / (-32) = 2
Since time cannot be negative in this situation, we discard the negative value (-0.25). Thus, the ball hits the ground after 2 seconds.
Therefore, it takes 2 seconds for the ball to hit the ground.
A. To determine if both t-intercepts of the graph have meaning in this situation, we need to understand what the t-intercepts represent. In a graph, the t-intercepts are the points where the graph intersects the t-axis, corresponding to the times when the height is zero.
To find the t-intercepts of the given function h(t) = -16t^2 + 28t + 8, we set h(t) = 0 and solve for t:
-16t^2 + 28t + 8 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. However, in this case, we can see that the equation does not factor easily.
Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / (2a), where a = -16, b = 28, and c = 8:
t = (-28 ± √(28^2 - 4*(-16)*8)) / (2*(-16))
t = (-28 ± √(784 + 512)) / (-32)
t = (-28 ± √1296) / (-32)
t = (-28 ± 36) / (-32)
From this, we have two t-intercepts:
t1 = (-28 + 36) / (-32) = 8 / (-32) = -0.25
t2 = (-28 - 36) / (-32) = -64 / (-32) = 2
Now, let's analyze the meaning of these t-intercepts in the given situation.
t = -0.25: This negative value implies that the ball was thrown before time zero, which is not possible in this context. Therefore, the t-intercept t = -0.25 does not have any meaningful interpretation in this situation.
t = 2: This positive value represents the time when the ball was caught or hit the ground. Since the problem states that no one caught the fly ball, the t-intercept t = 2 represents the time when the ball hit the ground.
B. From the analysis above, we found that the ball hits the ground at t = 2 seconds.