Question
A tennis ball is dropped from a certain height. Its height in feet is given by h, of, t, equals, minus, 16, t, squared, plus, 400h(t)=−16t
2
+400 where tt represents the time in seconds after launch. After how many seconds does the ball hit the ground?
2
+400 where tt represents the time in seconds after launch. After how many seconds does the ball hit the ground?
Answers
GPT-4o mini
To determine when the tennis ball hits the ground, we need to find the time \( t \) at which the height \( h(t) \) equals zero. The height of the ball is given by the equation:
\[
h(t) = -16t^2 + 400
\]
We set this equal to zero to find the time when the ball reaches the ground:
\[
-16t^2 + 400 = 0
\]
Now, we can solve for \( t \).
1. Rearranging the equation:
\[
-16t^2 = -400
\]
2. Dividing both sides by -16:
\[
t^2 = \frac{400}{16}
\]
\[
t^2 = 25
\]
3. Taking the square root of both sides:
\[
t = \sqrt{25}
\]
\[
t = 5
\]
Since time cannot be negative, we have \( t = 5 \) seconds.
Therefore, the tennis ball hits the ground after **5 seconds**.
\[
h(t) = -16t^2 + 400
\]
We set this equal to zero to find the time when the ball reaches the ground:
\[
-16t^2 + 400 = 0
\]
Now, we can solve for \( t \).
1. Rearranging the equation:
\[
-16t^2 = -400
\]
2. Dividing both sides by -16:
\[
t^2 = \frac{400}{16}
\]
\[
t^2 = 25
\]
3. Taking the square root of both sides:
\[
t = \sqrt{25}
\]
\[
t = 5
\]
Since time cannot be negative, we have \( t = 5 \) seconds.
Therefore, the tennis ball hits the ground after **5 seconds**.