Question
At baseball practice, Matt hit the ball into the outfield. The function h(t)=−16t2+32t+3h(t)=−16t2+32t+3 gives the height h(t)h(t) of the baseball in feet, tt seconds after it is hit. A) What is the value of h(1)h(1)? Response area B) What does h(1.74)h(1.74) represent in context? Response area C) Which value would not be valid in the context of this problem? Response area 193The time it took the ball to be 1.74 feet above the ground.h(2)h(2)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 height of the ball 1 second after it is hit.2.1h(0)h(0)h(−2)h(−2)
Answers
GPT-4o mini
To start, let's analyze the function given for the height of the baseball:
\[
h(t) = -16t^2 + 32t + 3
\]
### Part A: Calculate \( h(1) \)
To find \( h(1) \), substitute \( t = 1 \) into the function:
\[
h(1) = -16(1)^2 + 32(1) + 3
\]
\[
h(1) = -16 + 32 + 3
\]
\[
h(1) = 19
\]
So, **the value of \( h(1) \) is 19 feet.**
### Part B: What does \( h(1.74) \) represent in context?
The value \( h(1.74) \) represents the height of the baseball at exactly 1.74 seconds after it has been hit. This can be interpreted as how high off the ground the baseball is at that specific moment in time.
### Part C: Which value would not be valid in the context of this problem?
We assess the provided options:
1. **193**: This number does not refer to any defined context (not the height or time).
2. **The time it took the ball to be 1.74 feet above the ground.**: This is plausible, as it considers the height.
3. **\( h(2) \)**: The height of the ball at 2 seconds is valid (it can be calculated).
4. **The amount of time it takes for the ball to hit the ground.**: This is a valid measurement, though it requires solving for when height \( h(t) = 0 \).
5. **The height of the ball 1 second after it is hit.**: This is valid (it is \( h(1) = 19 \)).
6. **\( h(0) \)**: Valid as it means the initial height of the ball at \( t = 0 \).
7. **\( h(-2) \)**: This value would not be valid in context because a negative time (before the ball is hit) does not have a physical interpretation in this scenario.
Thus, **the value that would not be valid in the context of this problem is \( h(-2) \)**.
\[
h(t) = -16t^2 + 32t + 3
\]
### Part A: Calculate \( h(1) \)
To find \( h(1) \), substitute \( t = 1 \) into the function:
\[
h(1) = -16(1)^2 + 32(1) + 3
\]
\[
h(1) = -16 + 32 + 3
\]
\[
h(1) = 19
\]
So, **the value of \( h(1) \) is 19 feet.**
### Part B: What does \( h(1.74) \) represent in context?
The value \( h(1.74) \) represents the height of the baseball at exactly 1.74 seconds after it has been hit. This can be interpreted as how high off the ground the baseball is at that specific moment in time.
### Part C: Which value would not be valid in the context of this problem?
We assess the provided options:
1. **193**: This number does not refer to any defined context (not the height or time).
2. **The time it took the ball to be 1.74 feet above the ground.**: This is plausible, as it considers the height.
3. **\( h(2) \)**: The height of the ball at 2 seconds is valid (it can be calculated).
4. **The amount of time it takes for the ball to hit the ground.**: This is a valid measurement, though it requires solving for when height \( h(t) = 0 \).
5. **The height of the ball 1 second after it is hit.**: This is valid (it is \( h(1) = 19 \)).
6. **\( h(0) \)**: Valid as it means the initial height of the ball at \( t = 0 \).
7. **\( h(-2) \)**: This value would not be valid in context because a negative time (before the ball is hit) does not have a physical interpretation in this scenario.
Thus, **the value that would not be valid in the context of this problem is \( h(-2) \)**.