Primary Goal, Background and Assumptions: Your job now is to find the window of RPM that maximizes the average horsepower, subject to a constraint. The constraint is based on the fact that you can only put a finite number of gears in the car. Each gear determines a fixed ratio between RPM and speed. When you shift gears (assuming the speed doesn't change in the moment you're changing gears), the relative ratio between the gears will determine the ratio between the new and old RPMs. For the purposes of this problem, let us assume you are shifting between gears with a relative ratio of 1.5. This means the ratio of the RPMs must also be 1.5.
To make sure you understand the goal, background and assumptions, suppose you are looking at the graph and a classmate just walked up to see what you are doing. Your classmate doesn't understand the situation yet and asks the following misguided questions. How would you answer in order to set your classmate straight and explain what you're doing?
Link to the graph: pics on Flickr - Photo Sharing!
Question 1: "If we're trying to maximize it, why don't we make the window as big as possible (e.g. 0 to 8000 RPM)?"
Question 2: "If the peak gives us the best efficiency, why don't we just make the window be the peak itself and stay there the whole time?"