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# physics/Calculus help?

You have been hired as a mathematical analyst for a major car manufacturer. A “cruise control” system is to be designed for a mid-size car. The problem is first broken down into two parts: designing a system to convert real speed into recorded speed, and designing a mechanical apparatus that either slows down or speed up the car depending upon its recorded speed and a “speed set.” Your job is to solve the first part of the problem. The solution to the problem of converting real speed into recorded speed is again split into two stages. In the first stage, a metal pin is secured to the inside of the hub of the right front wheel of the car. This pin registers one unit on a counter with each revolution of the wheel. The counter is connected to a timer and a transmitter. In the second stage, the transmitter transmits the total count(i.e. number of revolutions of the wheel) to a “reader” each second (hence the need for a timer), and then the counter is cleared (i.e. set back to zero). The reader converts each such count into a recorded speed and displays that speed on a digital speedometer. The hub of the wheel has a radius of seven inches. Furthermore, the distance from the center of the wheel to the ground is twelve inches.

1. Give a rule for a function which has as its input (domain) the real speed of the car and which as its output(range) the count that would appear on the counter each second for a car traveling at that speed. Limit the domain of your graph to the interval [50,60]

2. Give a rule for a function that has as its input the count per second that could be transmitted by the transmitter and which has as its output the recorded speed of the car.

3. Explain how the function in (1) and (2) should be combined to give a function that has as its input the real speed of the car and as its output the recorded speed of the car. Carefully, draw the graph of this function over the interval [50,60].

4. Assume now that the cruise control device is about to be constructed. You, however, can see from examining your results thus far that there might be a problem when the control is “set” to 55 mph. Explain.

5. Two design alterations have been suggested to make the system better. One suggestion is to add additional equally spaced pins around the hub of the wheel. The formula the reader uses to convert the transmitted count into a recorded speed would have to be changed, of course. The second suggestion would be to transmit the count every 0.5 seconds instead of every second. Would you recommend implementing either or both of these suggestions? Why? Are there other improvements that you would suggest

6. Suppose that the car manufacturer now wishes to devise an instrument that could keep track of the total distance traveled while the cruise control is activated. How would you suggest that this be done?

I think it has something to do with torque and revolutions. Speed of car=50 miles/ hr —> #revolutions/sec??