Question
Destiny fills her truck with gas. The truck has some gas in it, and Destiny fills it at a constant rate each minute. Write an equation that represents this situation. Then use the equation to make a prediction.
This graph represents the linear relationship between the time in minutes and the gallons of gas in the truck. For each minute, the gallons of gas increases at a constant rate. What do the points (2, 50) and (4, 80) represent?
x
y
(2, 50)(4, 80)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Time (minutes)
Gallons of gas
The point (2, 50) means that after
minutes, there are
gallons of gas in the truck.
The point (4, 80) means that after
minutes, there are
gallons of gas in the truck.
This graph represents the linear relationship between the time in minutes and the gallons of gas in the truck. For each minute, the gallons of gas increases at a constant rate. What do the points (2, 50) and (4, 80) represent?
x
y
(2, 50)(4, 80)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Time (minutes)
Gallons of gas
The point (2, 50) means that after
minutes, there are
gallons of gas in the truck.
The point (4, 80) means that after
minutes, there are
gallons of gas in the truck.
Answers
GPT-4o mini
To represent the situation mathematically, we can define a linear equation in the form:
\[ y = mx + b \]
where:
- \( y \) is the total gallons of gas in the truck,
- \( x \) is the time in minutes,
- \( m \) is the rate of increase of gas per minute,
- \( b \) is the initial amount of gas already in the truck.
From the points given, (2, 50) and (4, 80), we can find the slope (\( m \)) of the line, which is the rate at which gas is being added.
The slope \( m \) can be calculated as follows:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{80 - 50}{4 - 2} = \frac{30}{2} = 15
\]
This means Destiny is adding gas at a rate of 15 gallons per minute.
Assuming that the truck started with a certain initial amount of gas (\( b \)), we can now use one of the points to solve for \( b \). We'll use the point (2, 50).
\[
50 = 15(2) + b \\
50 = 30 + b \\
b = 50 - 30 = 20
\]
Now we have all the information to write the equation:
\[
y = 15x + 20
\]
Now, regarding the points (2, 50) and (4, 80):
- The point (2, 50) means that **after 2 minutes**, there are **50 gallons of gas in the truck.**
- The point (4, 80) means that **after 4 minutes**, there are **80 gallons of gas in the truck.**
To make a prediction, we can use the equation to find the amount of gas after any given time. For example, if we want to know how much gas there is after 5 minutes:
\[
y = 15(5) + 20 = 75 + 20 = 95
\]
So, after 5 minutes, there will be **95 gallons of gas in the truck.**
\[ y = mx + b \]
where:
- \( y \) is the total gallons of gas in the truck,
- \( x \) is the time in minutes,
- \( m \) is the rate of increase of gas per minute,
- \( b \) is the initial amount of gas already in the truck.
From the points given, (2, 50) and (4, 80), we can find the slope (\( m \)) of the line, which is the rate at which gas is being added.
The slope \( m \) can be calculated as follows:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{80 - 50}{4 - 2} = \frac{30}{2} = 15
\]
This means Destiny is adding gas at a rate of 15 gallons per minute.
Assuming that the truck started with a certain initial amount of gas (\( b \)), we can now use one of the points to solve for \( b \). We'll use the point (2, 50).
\[
50 = 15(2) + b \\
50 = 30 + b \\
b = 50 - 30 = 20
\]
Now we have all the information to write the equation:
\[
y = 15x + 20
\]
Now, regarding the points (2, 50) and (4, 80):
- The point (2, 50) means that **after 2 minutes**, there are **50 gallons of gas in the truck.**
- The point (4, 80) means that **after 4 minutes**, there are **80 gallons of gas in the truck.**
To make a prediction, we can use the equation to find the amount of gas after any given time. For example, if we want to know how much gas there is after 5 minutes:
\[
y = 15(5) + 20 = 75 + 20 = 95
\]
So, after 5 minutes, there will be **95 gallons of gas in the truck.**