To analyze the situation provided, we need to derive the equation representing the linear relationship between the time (in minutes) and the gallons of gas in the bus.
First, let's understand the points given:
- 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.
Step 1: Calculate the slope (m)
The slope \( m \) can be calculated using the formula: \[ m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \( (4, 75) \) and \( (14, 175) \):
- \( (x_1, y_1) = (4, 75) \)
- \( (x_2, y_2) = (14, 175) \)
Plugging in the values:
\[ m = \frac{175 - 75}{14 - 4} = \frac{100}{10} = 10 \]
This means that for each minute, the gallons of gas in the bus increases by 10 gallons.
Step 2: Write the equation in the form \( y = mx + b \)
Now that we have the slope (\( m = 10 \)), we need to find the y-intercept (\( b \)). We can use one of the points to find \( b \). Let's use the point \( (4, 75) \):
\[ y = mx + b \quad \Rightarrow \quad 75 = 10(4) + b \] \[ 75 = 40 + b \] \[ b = 75 - 40 = 35 \]
Final Equation
Now we have both \( m \) and \( b \), so the equation that represents the situation is:
\[ y = 10x + 35 \]
where:
- \( y \) is the number of gallons of gas in the bus,
- \( x \) is the time in minutes.
Predictions
Using this equation, we can predict how many gallons of gas will be in the bus after any given number of minutes. For example:
- To find the amount of gas after 20 minutes: \[ y = 10(20) + 35 = 200 + 35 = 235 \text{ gallons} \]
Summary
- Slope \( m \): 10 (indicating 10 gallons of gas added per minute)
- Equation: \( y = 10x + 35 \)
- Interpretation of Points:
- (4, 75): After 4 minutes, there are 75 gallons of gas in the bus.
- (14, 175): After 14 minutes, there are 175 gallons of gas in the bus.