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Problem 70 Hard Difficulty

The graph shows the fuel consumption of $ c $ of a car (measured in gallons per hour) as a function of the speed $ v $ of the car. At very low speeds the engine runs inefficiency, so initially $ c $ decreases as the speed increases. But at high speeds the fuel consumption increases. You can see that $ c(V) $ is minimized for this car when $ v \approx 30 $ mi/h. However, for fuel efficiency, what must be minimized is not the consumption in gallons per hour but rather the fuel consumption in gallons per mile. Let's call this consumption $ G $. Using the graph, estimate the speed at which $ G $ has its minimum value.


$53 \mathrm{mi} / \mathrm{hr}$


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Ig N.

April 5, 2021

The graphs shows the fuel consumption of a car (measured in gallons per hour) as a function of the speed of a car. At very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. But at high speeds the fuel consumption i

Video Transcript

All right, We've got a question here that describes fuel consumption of a car. See as a function of speed of a car. Now very low speeds. It runs inefficiently. Excuse me on. See? Ah. Initially see, decreased at speed increases the high speeds. Fuel consumption increases on. I'm not gonna draw. Well, I gotta draw the graph here. Just give you a better visual depiction. You could see it in the textbook as well. Well, the 60 there. You know, something that looks like that there's something that looks kind of like a parabola and were asked to calculate using the graft estimate at what speed G has its minimum value. It's a G is consumption. And what speed would jihad its minimum consumption? It's the first one to create equation with, uh, G. That includes C and B. So, first of all, we know that G can be written as the fuel consumption. Excuse me. The quantity of the fuel over the distance and miles. Okay. And if you do this per mile, you could write fuel over time, distance over time. And we know that fuel consumption it's the same thing as fuel over time. So we could denote fuel consumption. See, we know distance over time it's the same thing as well. Awesome. All right, now, if we took the derivative change in G with respect to be come on. Yeah, And we said the change in G over TV to be equal to zero because we're trying to calculate, um, what our speed would be when our G is at its lowest. Then we could get this to come out to be the do you see over TV by the sea C is equal to bt seeding. Okay, so when this it is true, you have your, uh, this v here, you would have your smallest she guys. All right now, looking at the graph, we can create a relationship between C and B to help fill in these variables. So we would look at our parabola. We could write down equation. That's a C minus one. You go to M B minus 30. We're told that it is minimized when these approximately 30 squared here. If we know that would be is he put a zero. She is equal to 1.5 we can calculate for M and M will come out when you do the math there? 1800. Okay, so and if we write our see equation, we have 1800. The sorry am is equal to one over 1800. Get the math wrong there. I want you plug all those numbers in. We should get them equal to one over 1800. Yeah, B minus 30 squared plus one. But if you take the derivative of see, you could get D. C. D V. And the reason why we're doing this so that we can calculate our d c TV here and R c. And then we can calculate what RV would be when G is at its been. Yes. We're gonna take the derivative C with respect to velocity and that will come out to be one over 1800 times, two times the minus 30. All right, now, you can substitute our D c D v or this equation and our or C this equation. You can substitute those two equations into this and software V because now we have everything in terms of the So we would say one over 1800 multiplied by the minus 30 squared plus one equal to yeah, one over 1800 but by to the minus 30 times V. All right, then, when you go ahead and do all the math and solve for your V, you'll get your V to be equal to the square root. 2700, which is approximately 57 miles Brower. All right, so we can confirm that the speed would be 57 MPH for when g, the fuel consumption fuel consumption is at its lowest. All right, well, thank you so much for watching. I hope that clarifies the question.