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Calculus/Problem to solve

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Expert: - 5/18/2009


Question
Hi Scotto :)
I dont know how to calculate the following:
To calculate how many revolutions a tape has to go around a shaft of
diameter  4 cm if the tape has a thickness of 4 mm and when fully extended
has a length of 120 cm.  I do not know which formulas to use and do not
know how to solve calculus formulas. Plz help !!


Answer
One revolution is 2πr, and in this case r starts out as 4.
That means the first revolution is 8π.

On the next revolution, the radius is 4.4, so a revolution is 4.4π.

Basically, the length of the tape used after n revolutions is
4n + 0.4Σk where k goes from 0 to n-1.

The Σk for k going from 1 to n is n(n+1)/2.

That sum needs to be adjst to go from 0 to n-1 instead of 1 to n.

Rather than worrying about that,
just use (n+1)n/2 and subtract off 0.4n.

So the total length is [(4-0.4)n + 0.4Σk, k=1 to n]
= 3.6n + 0.4n(n+1)/2 = 3.6n + 0.2nČ + 0.2n = 0.2nČ + 3.8n.

We want to know when the total runs out, so subtract off 120.  
The result is 0.2nČ + 3.8n - 120.

All we have to do is use the quadratic equation to solve.
That is, (-b ± √(bČ - 4ac))/2a where a = 0.2, b = 3.8, and c = -120.
In this case, the ± will only be +, since a negative amount makes no sense.

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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Experience in the area: I have tutored students in all areas of mathematics for over 25 years. Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors. Awards and Honors: I have passed Actuarial tests 100, 110, and 135.

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