To determine the correct equation that models the amount of gas left in the tank as Phan drives, and to find out how many gallons of gasoline it takes to fill her tank, let's analyze the given data and derive the appropriate linear relationship.
Given the data points:
- After 3 hours of driving, there are 12 gallons of gas left.
- After 5 hours of driving, there are 8 gallons of gas left.
- After 7 hours of driving, there are 4 gallons of gas left.
- After 9 hours of driving, there are 0 gallons of gas left.
We can see from the data that the relationship between the number of hours spent driving ([tex]\(h\)[/tex]) and the amount of gas left ([tex]\(g\)[/tex]) is linear. Typically, this relationship can be represented by a linear equation of the form:
[tex]\[ g = mh + b \][/tex]
Where:
- [tex]\(m\)[/tex] is the slope (rate of gas consumption per hour).
- [tex]\(b\)[/tex] is the y-intercept (amount of gas when [tex]\(h = 0\)[/tex]).
To determine this linear relationship:
1. The slope [tex]\(m\)[/tex] can be calculated as the change in the amount of gas divided by the change in hours. Observing the changes:
- From 3 to 5 hours: (8 - 12) / (5 - 3) = -4 / 2 = -2 gallons/hour
- From 5 to 7 hours: (4 - 8) / (7 - 5) = -4 / 2 = -2 gallons/hour
- From 7 to 9 hours: (0 - 4) / (9 - 7) = -4 / 2 = -2 gallons/hour
This gives a consistent rate of gas consumption of -2 gallons per hour.
2. To find the y-intercept [tex]\(b\)[/tex], we can use one of the points. Let's use the point (5, 8):
[tex]\[ 8 = -2(5) + b \][/tex]
[tex]\[ 8 = -10 + b \][/tex]
[tex]\[ b = 18 \][/tex]
Therefore, the equation modeling the amount of gas left is:
[tex]\[ g = 18 - 2h \][/tex]
This equation reflects that the initial amount of gas in the tank (when [tex]\(h = 0\)[/tex]) is 18 gallons.
Thus, the correct choices from the given options are:
[tex]\[ g = 18 - 2h; 18 \text{ gallons} \][/tex]
Phan's car uses the equation [tex]\(g = 18 - 2h\)[/tex] to model the gas consumption rate, and it takes 18 gallons to fill her tank.
