# Optimization Q1

Hi,

I would like help to solve the following question.

Q: The graph shows the fuel consumption 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 inefficiently, 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~30mi/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. URL for graph is listed below.

http://img85.imageshack.us/img85/3246/imgtmp.pdf [Broken]

Thank you.

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## Answers and Replies

cronxeh
Gold Member
So your c(v) is in gallons/hr, and your v is the variable here. You want to find G = c(v)/v = gallons/hr / (miles/hr) = gallons/mile. Now find the minimum of G

c(v)/v = G
G/dv = (c'(v)*v - c(v)*v')/(v^2)

Find (c'(v)*v - c(v))/(v^2) = 0

If you give me the rest of that graph I can do curve fitting and perhaps get you the c(v) function and the solution to this

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Thank you. I will try that.