Question
A pool has some initial amount of water in it. Then it starts being filled so the water level rises at a rate of
\[6\] centimeters per minute. After
\[20\] minutes, the water level is
\[220\] centimeters.
Graph the relationship between the pool's water level (in centimeters) and time (in minutes).
\[6\] centimeters per minute. After
\[20\] minutes, the water level is
\[220\] centimeters.
Graph the relationship between the pool's water level (in centimeters) and time (in minutes).
Answers
GPT-4o mini
To graph the relationship between the pool's water level (in centimeters) and time (in minutes), we can start by determining the initial water level in the pool before the filling begins.
Given:
- The rate of rise in water level = \(6\) centimeters per minute
- Time duration of filling = \(20\) minutes
- Water level after \(20\) minutes = \(220\) centimeters
We can calculate the rise in water level after \(20\) minutes by using the rate of filling:
\[
\text{Rise in water level} = \text{rate} \times \text{time} = 6 \, \text{cm/min} \times 20 \, \text{min} = 120 \, \text{cm}
\]
Now, let’s calculate the initial water level in the pool:
\[
\text{Initial water level} = \text{Water level after filling} - \text{Rise in water level} = 220 \, \text{cm} - 120 \, \text{cm} = 100 \, \text{cm}
\]
Now we have the initial water level and the rate of rise. We can express the relationship mathematically as follows:
Let \( L(t) \) represent the water level in centimeters at time \( t \) minutes.
\[
L(t) = 100 + 6t
\]
Now we can plot this linear relationship on a graph where:
- \( x \)-axis (horizontal) represents time in minutes (\( t \))
- \( y \)-axis (vertical) represents water level in centimeters (\( L(t) \))
### Key Points for the Graph:
1. When \( t = 0 \):
\[
L(0) = 100 + 6 \cdot 0 = 100 \, \text{cm}
\]
Point: \( (0, 100) \)
2. When \( t = 20 \):
\[
L(20) = 100 + 6 \cdot 20 = 220 \, \text{cm}
\]
Point: \( (20, 220) \)
### Additional Points:
3. When \( t = 10 \):
\[
L(10) = 100 + 6 \cdot 10 = 160 \, \text{cm}
\]
Point: \( (10, 160) \)
### Graphing the Line:
- Connect the points \( (0, 100) \), \( (10, 160) \), and \( (20, 220) \) with a straight line.
- The line will have a slope of \( 6 \) degrees (rise/run) indicating that for every minute that passes, the water level rises by \( 6 \) centimeters.
### Conclusion:
The graph is a straight line starting from \( (0, 100) \) and ending at \( (20, 220) \), with the equation \( L(t) = 100 + 6t \) representing the relationship between water level and time.
Given:
- The rate of rise in water level = \(6\) centimeters per minute
- Time duration of filling = \(20\) minutes
- Water level after \(20\) minutes = \(220\) centimeters
We can calculate the rise in water level after \(20\) minutes by using the rate of filling:
\[
\text{Rise in water level} = \text{rate} \times \text{time} = 6 \, \text{cm/min} \times 20 \, \text{min} = 120 \, \text{cm}
\]
Now, let’s calculate the initial water level in the pool:
\[
\text{Initial water level} = \text{Water level after filling} - \text{Rise in water level} = 220 \, \text{cm} - 120 \, \text{cm} = 100 \, \text{cm}
\]
Now we have the initial water level and the rate of rise. We can express the relationship mathematically as follows:
Let \( L(t) \) represent the water level in centimeters at time \( t \) minutes.
\[
L(t) = 100 + 6t
\]
Now we can plot this linear relationship on a graph where:
- \( x \)-axis (horizontal) represents time in minutes (\( t \))
- \( y \)-axis (vertical) represents water level in centimeters (\( L(t) \))
### Key Points for the Graph:
1. When \( t = 0 \):
\[
L(0) = 100 + 6 \cdot 0 = 100 \, \text{cm}
\]
Point: \( (0, 100) \)
2. When \( t = 20 \):
\[
L(20) = 100 + 6 \cdot 20 = 220 \, \text{cm}
\]
Point: \( (20, 220) \)
### Additional Points:
3. When \( t = 10 \):
\[
L(10) = 100 + 6 \cdot 10 = 160 \, \text{cm}
\]
Point: \( (10, 160) \)
### Graphing the Line:
- Connect the points \( (0, 100) \), \( (10, 160) \), and \( (20, 220) \) with a straight line.
- The line will have a slope of \( 6 \) degrees (rise/run) indicating that for every minute that passes, the water level rises by \( 6 \) centimeters.
### Conclusion:
The graph is a straight line starting from \( (0, 100) \) and ending at \( (20, 220) \), with the equation \( L(t) = 100 + 6t \) representing the relationship between water level and time.