Let's start by formulating an equation for the situation described, where \( x \) is the time in minutes and \( y \) is the gallons of gas in the bus.
Given two points on the graph: \( (4, 75) \) and \( (14, 175) \), we can calculate the rate of change (slope) between these two points first:
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Calculate the slope (rate of change): \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{175 - 75}{14 - 4} = \frac{100}{10} = 10 \] This means that the bus is filled at a rate of 10 gallons per minute.
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Use one of the points to write the equation: We can use the slope-intercept form of the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Using the point \( (4, 75) \): \[ 75 = 10(4) + b \] \[ 75 = 40 + b \] \[ b = 75 - 40 = 35 \] The equation that represents the relationship between time and gallons of gas is: \[ y = 10x + 35 \]
Making Predictions
To predict the amount of gas at any time \( x \), we substitute \( x \) into the equation. For example, after 5 minutes: \[ y = 10(5) + 35 = 50 + 35 = 85 \] So, after 5 minutes, there will be 85 gallons of gas in the bus.
Interpret the Points
Now, referencing the points:
- 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.