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.
5 gallons = 220 miles
Let x = additional miles traveled
220/5 = x/10
I'll leave the explanations up to you.
To model this relationship with a linear equation, let's first define our variables:
Let's assume:
- The number of miles traveled is represented by 'x'
- The remaining gallons of gasoline is represented by 'y'
Now, let's analyze the given information:
- The salesman starts with 15 gallons of gasoline in his tank, which means when x = 0 (no miles traveled), y = 15 (gallons of gasoline remaining).
- After traveling 220 miles, he has 10 gallons of gasoline in his tank, which means when x = 220 (miles traveled), y = 10 (gallons of gasoline remaining).
We can use these two points to find the equation of the line in slope-intercept form: y = mx + b.
First, let's find the slope (m):
m = (y2 - y1) / (x2 - x1)
m = (10 - 15) / (220 - 0)
m = -5 / 220
m = -1/44
Now, let's find the y-intercept (b):
Using the point (0, 15) and the slope-intercept form, we can substitute the values to find the y-intercept:
15 = (-1/44) * 0 + b
15 = b
The linear equation relating the number of miles traveled (x) to the remaining gallons of gasoline (y) is:
y = (-1/44)x + 15
In this situation:
- The x-intercept represents the number of miles traveled when there is no gasoline remaining. In our equation, it would be the value of x when y = 0. So, we can solve the equation for x = 0:
0 = (-1/44)x + 15
(-1/44)x = -15
x = 660
Therefore, the x-intercept is 660, which means that when there are 660 miles traveled, there will be no gasoline remaining.
- The y-intercept represents the number of gallons of gasoline when no miles have been traveled. In our equation, it is the value of y when x = 0. So, substituting x = 0 in the equation:
y = (-1/44) * 0 + 15
y = 15
Therefore, the y-intercept is 15, which means that when no miles have been traveled, there will be 15 gallons of gasoline remaining.
- The slope (-1/44) represents the rate of change in the remaining gallons of gasoline per mile traveled. It tells us that for every 44 miles traveled, the gasoline decreases by 1 gallon.
To summarize:
- x-intercept: 660 (miles traveled when no gasoline remains)
- y-intercept: 15 (gallons of gasoline remaining when no miles have been traveled)
- slope (-1/44): The rate of change in the remaining gallons of gasoline per mile traveled.
To model the relationship between the number of miles traveled and the remaining gallons of gasoline, we can use the formula for a linear equation in slope-intercept form: y = mx + b. Let's break down the given information and incorporate it into this equation.
We are given that the salesman starts with 15 gallons of gasoline and can travel 220 miles before having 10 gallons of gasoline remaining. We can assign the number of miles traveled (x) as the independent variable and the remaining gallons of gasoline (y) as the dependent variable.
Using the slope-intercept form, we have the equation: y = mx + b.
In this scenario:
- The y-intercept (b) represents the starting point or the initial amount of gasoline in the salesman's tank. Here, the starting point is 15 gallons, so b = 15. In terms of the equation, it means that when no miles have been traveled (x = 0), the remaining gallons of gasoline are 15 (y = 15).
So, our equation becomes: y = mx + 15.
- The x-intercept represents the point where the remaining gallons of gasoline become zero, indicating that the salesman has run out of fuel. To find the x-intercept, we set y = 0 and solve for x. In this case, it represents the number of miles the salesman can travel before he runs out of gasoline.
0 = mx + 15
mx = -15
x = -15/m
The negative sign indicates that the x-intercept is the point before the salesman starts his trip, so we discard it in this context.
- The slope (m) represents the rate at which the remaining gallons of gasoline change with respect to the number of miles traveled. As the salesman travels, the remaining gallons decrease. To calculate the slope, we need to find the change in y divided by the change in x.
slope (m) = (y2 - y1) / (x2 - x1)
Given that the salesman starts with 15 gallons of gasoline and after traveling 220 miles, has 10 gallons remaining, we can substitute these values into the slope formula:
m = (10 - 15) / (220 - 0)
m = -5 / 220
m = -1/44
Therefore, the equation representing this scenario is y = (-1/44)x + 15. The x-intercept is not significant in this context, while the y-intercept represents the starting point (15 gallons) and the slope represents the rate of fuel consumption (-1/44 gallons per mile).