Question
Patty has been watching her rain gauge during a heavy storm. She made her first observation when she woke up in the morning and has continued to watch throughout the day.
There is a linear relationship between the time, in hours, since Patty's first observation, and the total amount of rainfall, in inches.
There is a linear relationship between the time, in hours, since Patty's first observation, and the total amount of rainfall, in inches.
Answers
GPT-4o mini
To describe Patty's observations of rainfall in relation to time, we can formulate a linear equation based on the given information.
Let's define:
- \( t \): the time in hours since Patty's first observation,
- \( R(t) \): the total amount of rainfall in inches at time \( t \).
Since there is a linear relationship, we can express this relationship in the form of a linear equation:
\[
R(t) = mt + b
\]
where:
- \( m \) is the slope of the line, representing the rate of rainfall in inches per hour,
- \( b \) is the y-intercept, representing the initial amount of rainfall (the amount in the gauge at the time of Patty's first observation).
Assuming Patty made her first observation at \( t = 0 \), then \( b \) would be the rainfall measurement at that moment. To analyze the rain accumulation:
- If Patty recorded specific amounts of rainfall at certain times, you could substitute those values into the equation to determine \( m \) and \( b \).
For example:
1. If Patty recorded 0 inches of rain at \( t = 0 \) (her first observation),
2. After 2 hours, she recorded 1 inch of rain.
3. After 4 hours, she recorded 3 inches of rain.
Using these points, you can find \( m \) and \( b \):
- From the first observation \( R(0) = 0 \) inches, so \( b = 0 \).
- Using the points (2, 1) and (4, 3), we can find the slope \( m \):
The slope \( m \) can be calculated as:
\[
m = \frac{R(4) - R(2)}{4 - 2} = \frac{3 - 1}{4 - 2} = \frac{2}{2} = 1 \text{ inch/hour}
\]
Now, plugging \( m \) and \( b \) into the linear equation, we have:
\[
R(t) = 1t + 0 \\
R(t) = t
\]
This means that for every hour that passes, the total rainfall increases by 1 inch.
If you have specific time and rainfall data, you can repeat this process to identify the linear relationship. Feel free to share any specific observations, and I can help you determine \( m \) and \( b \) more accurately!
Let's define:
- \( t \): the time in hours since Patty's first observation,
- \( R(t) \): the total amount of rainfall in inches at time \( t \).
Since there is a linear relationship, we can express this relationship in the form of a linear equation:
\[
R(t) = mt + b
\]
where:
- \( m \) is the slope of the line, representing the rate of rainfall in inches per hour,
- \( b \) is the y-intercept, representing the initial amount of rainfall (the amount in the gauge at the time of Patty's first observation).
Assuming Patty made her first observation at \( t = 0 \), then \( b \) would be the rainfall measurement at that moment. To analyze the rain accumulation:
- If Patty recorded specific amounts of rainfall at certain times, you could substitute those values into the equation to determine \( m \) and \( b \).
For example:
1. If Patty recorded 0 inches of rain at \( t = 0 \) (her first observation),
2. After 2 hours, she recorded 1 inch of rain.
3. After 4 hours, she recorded 3 inches of rain.
Using these points, you can find \( m \) and \( b \):
- From the first observation \( R(0) = 0 \) inches, so \( b = 0 \).
- Using the points (2, 1) and (4, 3), we can find the slope \( m \):
The slope \( m \) can be calculated as:
\[
m = \frac{R(4) - R(2)}{4 - 2} = \frac{3 - 1}{4 - 2} = \frac{2}{2} = 1 \text{ inch/hour}
\]
Now, plugging \( m \) and \( b \) into the linear equation, we have:
\[
R(t) = 1t + 0 \\
R(t) = t
\]
This means that for every hour that passes, the total rainfall increases by 1 inch.
If you have specific time and rainfall data, you can repeat this process to identify the linear relationship. Feel free to share any specific observations, and I can help you determine \( m \) and \( b \) more accurately!