Suppose you take a car trip, traveling east along a very long highway, starting at time t 0. Let x t be the number of gallons of gasoline used during the first t hours, and let st be the distance traveled in that time. Because you’re using very cheap gasoline that’s not good for the engine, your fuel efficiency (measured in miles/gallon) decreases as fuel is consumed. The graph of fuel efficiency as a function of gasoline used is shown in the graph below.
You record the total fuel used at various times during the trip and make the graph below, showing amount of fuel used as a function of time.
Using only these two graphs, estimate your instantaneous velocity 15 hours after the start of the trip. Explain how you arrived at that value
Given in the graphs.
A hint was given not to try and find formulas for the graphs
x t = number of gallons used for the first t hours
s(t) = distance traveled in that time
The Attempt at a Solution
So, for 15 hrs, according to the second graph. we can tell that the overall gallons of fuel used at t = 15 is x =9( I think). From that we can tell that the interval of fuel efficiency that we will use is (30, 25) as the efficiency is 25 at x = 9. However, if x t is the number of gallons used, then our solution gives us 15x, which contrasts with the second graph we're given. Honestly, I'm just not sure how to proceed with the question. All the related rates problems I've done so far have provided functions instead of graphs, and here I can't think of a way to find the relationship between miles per gallon and the instantaneous velocity.
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