Question
Mike threw a rock off a cliff that is roughly 10 meters above a lake. It can be modeled by the equation h = -4.9t^2+10 where h is the height in meters and t is the time in seconds If you set up a table to determine when the rock hits the water. What would be a reasonable scale for the range
Answers
Answered by
GPT-4o mini
To determine when the rock hits the water using the equation \( h = -4.9t^2 + 10 \), we need to find the time \( t \) when \( h = 0 \) (the height of the water).
Setting up the equation:
\[
0 = -4.9t^2 + 10
\]
Rearranging gives:
\[
4.9t^2 = 10
\]
Dividing both sides by 4.9:
\[
t^2 = \frac{10}{4.9} \approx 2.0408
\]
Then taking the square root:
\[
t \approx \sqrt{2.0408} \approx 1.43 \text{ seconds}
\]
Now, to set up a table to determine when the rock hits the water, we can choose a scale for the time \( t \) that includes the time up to when the rock hits the water.
Since the rock hits the water at approximately \( t = 1.43 \) seconds, a reasonable scale for the range of time could be from \( 0 \) to about \( 2 \) seconds, allowing us to see the entire motion of the rock as it falls.
A possible table setup could include time intervals such as \( 0 \), \( 0.5 \), \( 1.0 \), \( 1.5 \), and \( 2.0 \) seconds.
Here's an example of how the table might look:
| Time (t) | Height (h) |
|----------|------------|
| 0.0 | 10.0 |
| 0.5 | 6.23 |
| 1.0 | 5.0 |
| 1.5 | 0.2 |
| 2.0 | -9.8 |
You can calculate the height at each of these time points using the given equation. This table will help visualize when the rock hits the water as well.
Setting up the equation:
\[
0 = -4.9t^2 + 10
\]
Rearranging gives:
\[
4.9t^2 = 10
\]
Dividing both sides by 4.9:
\[
t^2 = \frac{10}{4.9} \approx 2.0408
\]
Then taking the square root:
\[
t \approx \sqrt{2.0408} \approx 1.43 \text{ seconds}
\]
Now, to set up a table to determine when the rock hits the water, we can choose a scale for the time \( t \) that includes the time up to when the rock hits the water.
Since the rock hits the water at approximately \( t = 1.43 \) seconds, a reasonable scale for the range of time could be from \( 0 \) to about \( 2 \) seconds, allowing us to see the entire motion of the rock as it falls.
A possible table setup could include time intervals such as \( 0 \), \( 0.5 \), \( 1.0 \), \( 1.5 \), and \( 2.0 \) seconds.
Here's an example of how the table might look:
| Time (t) | Height (h) |
|----------|------------|
| 0.0 | 10.0 |
| 0.5 | 6.23 |
| 1.0 | 5.0 |
| 1.5 | 0.2 |
| 2.0 | -9.8 |
You can calculate the height at each of these time points using the given equation. This table will help visualize when the rock hits the water as well.