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Question Video: Analyzing a Graph to Find an Equation for the Most Environmentally Friendly Car Mathematics • Second Year of Secondary School

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The lines on the following graph show the amount of fuel remaining in the tank as a function of the distance covered since the tank was 60 L full for three different car models. Which of the following is the function describing the quantity of fuel in the tank of the most environmentally friendly car model represented in the above graph? [A] 𝑓(π‘₯) = 60 βˆ’ (3/40)π‘₯ [B] 𝑓(π‘₯) = 60 βˆ’ (1/10)π‘₯ [C] 𝑓(π‘₯) = 60 βˆ’ (1/15)π‘₯ [D] 𝑓(π‘₯) = (1/15)π‘₯ βˆ’ 60 [E] 𝑓(π‘₯) = (3/40)π‘₯ βˆ’ 60

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Video Transcript

The lines on the following graph show the amount of fuel remaining in the tank as a function of the distance covered since the tank was 60 liters full for three different car models. Which of the following is the function describing the quantity of fuel in the tank of the most environmentally friendly car model represented in the above graph? Option (A) 𝑓 of π‘₯ equals 60 minus three over 40 times π‘₯. Option (B) 𝑓 of π‘₯ equals 60 minus one over 10 times π‘₯. Option (C) 𝑓 of π‘₯ equals 60 minus one over 15 times π‘₯. Option (D) 𝑓 of π‘₯ equals one over 15 π‘₯ minus 60. Or is it option (E) 𝑓 of π‘₯ equals three over 40 π‘₯ minus 60?

In this question, we are given a graph showing the amount of fuel remaining in the tanks of three different models of car as a function of the distance traveled. We want to use this graph to identify the function describing the quantity of fuel remaining in the most environmentally friendly car after it has traveled π‘₯ kilometers. To do this, let’s start by considering which model of car will be the most environmentally friendly. We can assume that this will be the car that travels the furthest distance on the 60 liters of fuel.

Since the π‘₯-coordinates of points on the graph tell us the distance traveled and the 𝑦-coordinates tell us the amount of fuel remaining, we can note that the π‘₯-intercepts will tell us the distance traveled when there is no fuel remaining in the tank. We see that the black line has the largest π‘₯-intercept at 900, so it is the most fuel-efficient car model in this test. We want to find the function representing the black line in this graph. We can achieve this by recalling that a straight line has the equation 𝑦 equals π‘šπ‘₯ plus 𝑏, where π‘š is the slope of the line and 𝑏 is its 𝑦-intercept. This is the same as saying that 𝑓 of π‘₯ equals π‘šπ‘₯ plus 𝑏, so we just need to find the slope and 𝑦-intercept of the black line in the graph.

We can start with the 𝑦-intercept. We see that all three lines share the same 𝑦-intercept of 60, that is, the initial amount of fuel in the car models. We can find the slope by recalling it is the change in 𝑦 divided by the change in π‘₯. We see that as we travel between the intercepts on the black line, the 𝑦-value decreases from 60 to zero and the π‘₯-values increase from zero to 900. Therefore, the slope is equal to negative 60 over 900, which simplifies to give negative one over 15. If we substitute π‘š equals negative one over 15 and 𝑏 equals 60 into the equation of a line, we obtain 𝑓 of π‘₯ equals negative one over 15 π‘₯ plus 60. We see that this is the same as option (C) with the terms reversed.

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