Question
The graph models the swimming paths of two students, where x represents the time after a buzzer sounds and y represents height from the pool edge. One student dived into the pool from the edge of the pool 3 minutes before the buzzer went off. The second student dived into the pool from a board 4 feet above the ground right when the buzzer went off. Which of the following points can be used to best estimate the time when the divers are at the same depth? (1 point) 3x+y=4 and x + y = -3
Answers
GPT-4o mini
To determine the time when the two divers are at the same depth using the equations you provided, we need to analyze the equations step by step. The equations given are:
1. \( 3x + y = 4 \)
2. \( x + y = -3 \)
Where \( x \) represents time (in minutes) and \( y \) represents the height from the pool edge.
### Step 1: Solve the equations
1. From the second equation \( x + y = -3 \), we can express \( y \) in terms of \( x \):
\[
y = -x - 3
\]
2. Substitute \( y \) into the first equation \( 3x + y = 4 \):
\[
3x + (-x - 3) = 4
\]
\[
3x - x - 3 = 4
\]
\[
2x - 3 = 4
\]
\[
2x = 7
\]
\[
x = \frac{7}{2} = 3.5
\]
### Step 2: Substitute \( x \) back to find \( y \)
Now we can find \( y \):
\[
y = -\left(\frac{7}{2}\right) - 3 = -\frac{7}{2} - \frac{6}{2} = -\frac{13}{2} = -6.5
\]
### Conclusion
The point that best estimates the time when the divers are at the same depth is:
\[
\left(\frac{7}{2}, -\frac{13}{2}\right) \text{ or } (3.5, -6.5)
\]
This means at approximately 3.5 minutes after the buzzer, both divers are at the same height, which we calculated to be -6.5 feet indicating that they are under the water (depth).
Thus, the best estimate for the time when the two divers are at the same depth is \( x = 3.5 \) minutes after the buzzer, or \( 3 \) minutes and \( 30 \) seconds after the buzzer.
1. \( 3x + y = 4 \)
2. \( x + y = -3 \)
Where \( x \) represents time (in minutes) and \( y \) represents the height from the pool edge.
### Step 1: Solve the equations
1. From the second equation \( x + y = -3 \), we can express \( y \) in terms of \( x \):
\[
y = -x - 3
\]
2. Substitute \( y \) into the first equation \( 3x + y = 4 \):
\[
3x + (-x - 3) = 4
\]
\[
3x - x - 3 = 4
\]
\[
2x - 3 = 4
\]
\[
2x = 7
\]
\[
x = \frac{7}{2} = 3.5
\]
### Step 2: Substitute \( x \) back to find \( y \)
Now we can find \( y \):
\[
y = -\left(\frac{7}{2}\right) - 3 = -\frac{7}{2} - \frac{6}{2} = -\frac{13}{2} = -6.5
\]
### Conclusion
The point that best estimates the time when the divers are at the same depth is:
\[
\left(\frac{7}{2}, -\frac{13}{2}\right) \text{ or } (3.5, -6.5)
\]
This means at approximately 3.5 minutes after the buzzer, both divers are at the same height, which we calculated to be -6.5 feet indicating that they are under the water (depth).
Thus, the best estimate for the time when the two divers are at the same depth is \( x = 3.5 \) minutes after the buzzer, or \( 3 \) minutes and \( 30 \) seconds after the buzzer.