Question
At baseball practice, Matt hot the ball into the outfield. The function h(t) = - 16t^2 + 3 gives the height h (t) of the baseball in feet, t seconds after it is hit.
h (t)
19
3
2.1
h (-2)
The height of the ball 1 second after it is hit
The amount of time it takes for the ball to hit the ground
The height.
Of the ball 1.74 seconds after it is hit
The time it took the ball to be1.74 feet above the ground
h (2)
h (t)
19
3
2.1
h (-2)
The height of the ball 1 second after it is hit
The amount of time it takes for the ball to hit the ground
The height.
Of the ball 1.74 seconds after it is hit
The time it took the ball to be1.74 feet above the ground
h (2)
Answers
GPT-4o mini
The function given is \( h(t) = -16t^2 + 3 \), which models the height of the baseball in feet at time \( t \) seconds after it is hit. We will analyze the function to provide answers to each of your requests.
1. **Height of the ball 1 second after it is hit (t=1)**:
\[
h(1) = -16(1)^2 + 3 = -16 + 3 = -13 \text{ feet}
\]
The height is -13 feet. Since height cannot be negative, this means the ball has already hit the ground before the 1-second mark.
2. **The amount of time it takes for the ball to hit the ground**:
To find out when the ball hits the ground, we set \( h(t) = 0 \):
\[
-16t^2 + 3 = 0
\]
\[
16t^2 = 3
\]
\[
t^2 = \frac{3}{16}
\]
\[
t = \sqrt{\frac{3}{16}} = \frac{\sqrt{3}}{4} \approx 0.433 \text{ seconds}
\]
Thus, it takes approximately 0.433 seconds for the ball to hit the ground.
3. **Height of the ball 1.74 seconds after it is hit (t=1.74)**:
\[
h(1.74) = -16(1.74)^2 + 3
\]
\[
= -16(3.0276) + 3 \approx -48.4416 + 3 \approx -45.4416 \text{ feet}
\]
Again, since the height is negative, this indicates that the ball has already hit the ground before reaching 1.74 seconds.
4. **The time it took the ball to be 1.74 feet above the ground**:
Set \( h(t) = 1.74 \):
\[
-16t^2 + 3 = 1.74
\]
\[
-16t^2 = 1.74 - 3
\]
\[
-16t^2 = -1.26
\]
\[
t^2 = \frac{1.26}{16} \approx 0.07875
\]
\[
t \approx \sqrt{0.07875} \approx 0.28 \text{ seconds}
\]
Therefore, it took approximately 0.28 seconds for the ball to reach a height of 1.74 feet.
5. **The height at \( t = 2 \)**:
\[
h(2) = -16(2)^2 + 3
\]
\[
= -16(4) + 3 = -64 + 3 = -61 \text{ feet}
\]
Again, a negative height indicates the ball has hit the ground well before the 2-second mark.
In summary:
- \( h(1) \approx -13 \) feet
- Time to hit the ground: \( \approx 0.433 \) seconds
- \( h(1.74) \approx -45.44 \) feet
- Time to be \( 1.74 \) feet above the ground: \( \approx 0.28 \) seconds
- \( h(2) \approx -61 \) feet
1. **Height of the ball 1 second after it is hit (t=1)**:
\[
h(1) = -16(1)^2 + 3 = -16 + 3 = -13 \text{ feet}
\]
The height is -13 feet. Since height cannot be negative, this means the ball has already hit the ground before the 1-second mark.
2. **The amount of time it takes for the ball to hit the ground**:
To find out when the ball hits the ground, we set \( h(t) = 0 \):
\[
-16t^2 + 3 = 0
\]
\[
16t^2 = 3
\]
\[
t^2 = \frac{3}{16}
\]
\[
t = \sqrt{\frac{3}{16}} = \frac{\sqrt{3}}{4} \approx 0.433 \text{ seconds}
\]
Thus, it takes approximately 0.433 seconds for the ball to hit the ground.
3. **Height of the ball 1.74 seconds after it is hit (t=1.74)**:
\[
h(1.74) = -16(1.74)^2 + 3
\]
\[
= -16(3.0276) + 3 \approx -48.4416 + 3 \approx -45.4416 \text{ feet}
\]
Again, since the height is negative, this indicates that the ball has already hit the ground before reaching 1.74 seconds.
4. **The time it took the ball to be 1.74 feet above the ground**:
Set \( h(t) = 1.74 \):
\[
-16t^2 + 3 = 1.74
\]
\[
-16t^2 = 1.74 - 3
\]
\[
-16t^2 = -1.26
\]
\[
t^2 = \frac{1.26}{16} \approx 0.07875
\]
\[
t \approx \sqrt{0.07875} \approx 0.28 \text{ seconds}
\]
Therefore, it took approximately 0.28 seconds for the ball to reach a height of 1.74 feet.
5. **The height at \( t = 2 \)**:
\[
h(2) = -16(2)^2 + 3
\]
\[
= -16(4) + 3 = -64 + 3 = -61 \text{ feet}
\]
Again, a negative height indicates the ball has hit the ground well before the 2-second mark.
In summary:
- \( h(1) \approx -13 \) feet
- Time to hit the ground: \( \approx 0.433 \) seconds
- \( h(1.74) \approx -45.44 \) feet
- Time to be \( 1.74 \) feet above the ground: \( \approx 0.28 \) seconds
- \( h(2) \approx -61 \) feet