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Difficulty: MediumLinear Equations in Two Variables

A hybrid car's fuel tank has a capacity of 1212 gallons. When driving on a highway, the amount of fuel in the tank decreases at a constant rate. After driving for 1.51.5 hours, 9.69.6 gallons of fuel remain in the tank. If the amount of fuel in the tank, FF, in gallons, after driving for tt hours is modeled by a linear equation, what is the value of FF when t=4t = 4?

Answer: 5.6 gallons

Answer

The amount of fuel remaining in the tank after driving for 44 hours is 5.65.6 gallons.
The amount of fuel in the tank, FF, and the driving time, tt, share a linear relationship. The initial amount of fuel at t=0t = 0 is 1212 gallons, representing the vertical intercept. The fuel decreases at a constant rate, which is the slope of the line. Over a period of 1.51.5 hours, the amount of fuel decreases by 12−9.6=2.412 - 9.6 = 2.4 gallons. The rate of decrease is 2.41.5=1.6\frac{2.4}{1.5} = 1.6 gallons per hour, so the slope is −1.6-1.6. The linear model is F=−1.6t+12F = -1.6t + 12. Substituting t=4t = 4 into this equation yields F=−1.6(4)+12=−6.4+12=5.6F = -1.6(4) + 12 = -6.4 + 12 = 5.6 gallons.

Step-by-Step Solution

1
Calculate the constant rate of fuel consumption (the slope of the linear relationship).
The rate of consumption is 1.61.6 gallons per hour.
Since the fuel decreases at a constant rate, the change in fuel divided by the change in time gives the rate of consumption. In 1.51.5 hours, the fuel decreases from 1212 gallons to 9.69.6 gallons, which is a decrease of 12−9.6=2.412 - 9.6 = 2.4 gallons. Thus, the rate of consumption is 2.4 gallons1.5 hours=1.6\frac{2.4\text{ gallons}}{1.5\text{ hours}} = 1.6 gallons per hour.
2
Write the linear equation modeling the fuel remaining in the tank, FF, as a function of time, tt.
F=−1.6t+12F = -1.6t + 12
The initial amount of fuel when t=0t = 0 is 1212 gallons, which represents the vertical intercept (b=12b = 12). The fuel decreases at a constant rate of 1.61.6 gallons per hour, which represents a slope of m=−1.6m = -1.6.
3
Substitute t=4t = 4 into the linear equation to find the value of FF.
F=5.6F = 5.6
Evaluating the equation at t=4t = 4 yields F=−1.6(4)+12=−6.4+12=5.6F = -1.6(4) + 12 = -6.4 + 12 = 5.6.

Key Concept

Linear Equations in Two Variables
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Topics

Linear Equations in Two Variables

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