Let's represent the situation with an equation first. We know that the relationship between the time (in minutes) and the gallons of gas in the bus is linear, meaning it can be described by the equation:
\[ g(t) = mt + b \]
Where:
- \( g(t) \) is the number of gallons of gas in the bus after \( t \) minutes.
- \( m \) is the rate at which gas is added per minute.
- \( b \) is the initial amount of gas in the bus at \( t = 0 \).
From the points (4, 75) and (14, 175), we can derive the rate and the initial amount of gas.
- First, find the rate \( m \):
The change in gas over the change in time between the two points: \[ m = \frac{175 - 75}{14 - 4} = \frac{100}{10} = 10 \]
This indicates that Corey adds 10 gallons of gas each minute.
- Next, we can use one of the points to find the initial amount of gas \( b \). Let's use the point (4, 75).
Substituting \( m \) and the point (4, 75) into the equation: \[ 75 = 10(4) + b \] \[ 75 = 40 + b \] \[ b = 75 - 40 = 35 \]
Thus, the complete equation representing the number of gallons of gas in the bus after \( t \) minutes is:
\[ g(t) = 10t + 35 \]
Predictions using the equation:
To predict the number of gallons of gas after a specific time, we can plug in values of \( t \) into our equation.
Now, let's address the points (4, 75) and (14, 175):
- The point (4, 75) means that after 4 minutes, there are 75 gallons of gas in the bus.
- The point (14, 175) means that after 14 minutes, there are 175 gallons of gas in the bus.
To summarize:
- (4, 75) means that after 4 minutes, there are 75 gallons of gas in the bus.
- (14, 175) means that after 14 minutes, there are 175 gallons of gas in the bus.