A salesman starts a trip with 15 gallons of gasoline in his tank. After traveling 220 miles he has 10 gallons of gasoline in his tank. Model this relationship with a linear equation relating the number of miles you can travel to the remaining gallons of gasoline. Explain what the x-intercept, the y-intercept, and the slope represent in this situation.
he drives 220 mi on 5 gal ... 44 mi/gal
... slope is the car's MPG
miles on remaining fuel is the range of the car
... range is DEPENDENT on gallons of fuel
gallons on x-axis ... range on y-axis
no fuel means no range
... both intercepts are zero (the origin)
To model this relationship with a linear equation, we need to find the equation in the form of y = mx + b, where "y" represents the number of miles traveled, "x" represents the remaining gallons of gasoline, "m" is the slope, and "b" is the y-intercept.
Given that the salesman starts with 15 gallons of gasoline and travels 220 miles with 10 gallons remaining, we can find the slope:
The change in y is 220 - 0 = 220 (miles traveled)
The change in x is 15 - 10 = 5 (gallons of gasoline remaining)
Therefore, the slope (m) is 220 / 5 = 44.
Now, let's find the y-intercept (b). We know that when x = 0 (i.e., when the salesman has used all the gasoline), y = 220. So, b = 220.
Thus, the linear equation is y = 44x + 220.
Now, let's interpret the terms in the context of the situation:
1. x-intercept: In this scenario, the x-intercept is the number of remaining gallons of gasoline (x) when the salesman has traveled 0 miles (y = 0). To find the x-intercept, we set y = 0 in the equation:
0 = 44x + 220
44x = -220
x = -5
However, having negative remaining gallons of gasoline makes no sense in this context. So, we can conclude that there is no x-intercept in this situation.
2. y-intercept: In this scenario, the y-intercept is the number of miles traveled (y) when the salesman has used all the gasoline (x = 0). Substituting x = 0 into the equation:
y = 44(0) + 220
y = 220
Thus, the y-intercept represents the initial number of miles (220) the salesman can travel when he starts with a full tank (15 gallons) of gasoline.
3. Slope: In this situation, the slope (m) represents the rate at which the number of miles traveled changes with each gallon of gasoline consumed. The slope of 44 means that for every gallon of gasoline used, the salesman can travel 44 miles.
Therefore, in this scenario, the x-intercept is not applicable, the y-intercept represents the initial miles the salesman can travel, and the slope represents the mileage covered per gallon of gasoline.