Question
A ball was kicked from the ground with a velocity of 64 feet per second and followed the path modeled by the function: h(t)=−16t^2+64t
.(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The vertex of the rock's path indicates the rock took seconds to reach its maximum height of feet. Looking at the graph of this function, we can see the domain in context to this situation is 0 to 4 because the ball .
.(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The vertex of the rock's path indicates the rock took seconds to reach its maximum height of feet. Looking at the graph of this function, we can see the domain in context to this situation is 0 to 4 because the ball .
Answers
GPT-4o mini
To find the vertex of the function \( h(t) = -16t^2 + 64t \), we use the formula for the vertex of a quadratic function \( t = -\frac{b}{2a} \), where \( a \) is the coefficient of \( t^2 \) and \( b \) is the coefficient of \( t \).
In this case:
- \( a = -16 \)
- \( b = 64 \)
Now calculate the time at which the maximum height occurs:
\[
t = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2 \text{ seconds}
\]
Next, we find the maximum height by substituting \( t = 2 \) back into the height function:
\[
h(2) = -16(2^2) + 64(2) = -16(4) + 128 = -64 + 128 = 64 \text{ feet}
\]
Now, let's fill in the answers based on what we've calculated:
1. The vertex of the rock's path indicates the rock took **2** seconds to reach its maximum height of **64** feet.
2. The ball is projected into the air and will hit the ground after **4** seconds based on the equation \( h(t) = 0 \). Analyzing the function, we can also derive that \( h(t) = 0 \) when \( t = 0 \) and \( t = 4 \).
Therefore, "Looking at the graph of this function, we can see the domain in context to this situation is 0 to 4 because the ball **returns to the ground**."
So the final answers would be:
- **2** seconds, **64** feet
- returns to the ground
In this case:
- \( a = -16 \)
- \( b = 64 \)
Now calculate the time at which the maximum height occurs:
\[
t = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2 \text{ seconds}
\]
Next, we find the maximum height by substituting \( t = 2 \) back into the height function:
\[
h(2) = -16(2^2) + 64(2) = -16(4) + 128 = -64 + 128 = 64 \text{ feet}
\]
Now, let's fill in the answers based on what we've calculated:
1. The vertex of the rock's path indicates the rock took **2** seconds to reach its maximum height of **64** feet.
2. The ball is projected into the air and will hit the ground after **4** seconds based on the equation \( h(t) = 0 \). Analyzing the function, we can also derive that \( h(t) = 0 \) when \( t = 0 \) and \( t = 4 \).
Therefore, "Looking at the graph of this function, we can see the domain in context to this situation is 0 to 4 because the ball **returns to the ground**."
So the final answers would be:
- **2** seconds, **64** feet
- returns to the ground