The Zepeda family filled the gasoline tank of their minivan at the beginning of a long trip. The relationship between the number of miles driven and the number of gallons remaining in the tank can be modeled with a linear function. When they had driven 120 miles, there were 18 gallons of gasoline in the tank. When they had driven 300 miles, only 6 gallons were left in the tank. How many miles could the Zepedas drive before running out of gasoline?

Question

The Zepeda family filled the gasoline tank of their minivan at the beginning of a long trip. The relationship between the number of miles driven and the number of gallons remaining in the tank can be modeled with a linear function. When they had driven 120 miles, there were 18 gallons of gasoline in the tank. When they had driven 300 miles, only 6 gallons were left in the tank. How many miles could the Zepedas drive before running out of gasoline?

Answer 1

If we model the situation as

y = mx+b

then we have

120m+b = 18
300m+b = 6
Subtract to get
-180m = 12
m = -1/15
So, b = 26

y = -1/15 x + 26

y=0 when x = 390

Answer 2

To find out how many miles the Zepeda family can drive before running out of gasoline, we need to determine the linear function that models the relationship between the number of miles driven and the number of gallons remaining in the tank.

Let's first find the slope of the linear function. The slope represents the rate at which the number of gallons decreases per mile driven. We can calculate the slope using the formula:

slope = (change in gallons) / (change in miles)

The change in gallons is 18 - 6 = 12 gallons, and the change in miles is 300 - 120 = 180 miles.

slope = 12 gallons / 180 miles = 1/15 gallons per mile

Now that we have the slope, we can use the point-slope form of a linear equation to find the equation for the function. We can choose either of the given points (120, 18) or (300, 6).

Using the point-slope form:

y - y1 = m(x - x1)

Let's use the point (300, 6):

y - 6 = (1/15)(x - 300)

Now, let's rearrange this equation to the slope-intercept form (y = mx + b), where b is the y-intercept:

y = (1/15)x - 4 + 6

Simplifying the equation:

y = (1/15)x + 2

Since the y-coordinate represents the number of gallons remaining in the tank and we want to know the number of miles they can drive before the tank runs out of gasoline, we need to find the x-coordinate (miles) when y (gallons) is zero.

0 = (1/15)x + 2

Subtracting 2 from both sides of the equation:

(1/15)x = -2

Now, multiply both sides by 15 to isolate x:

15 * (1/15)x = -2 * 15

x = -30

Since negative miles don't make sense in this context, it implies that the Zepeda family will run out of gasoline before driving any miles. Therefore, we can conclude that they cannot drive any miles before running out of gasoline based on the given information.