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January 15th, 2019, 04:56 AM   #1
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Math Focus: Calculus
Question How to do the computation of a spiral length in a disk using college algebra?

I have found this riddle in my book and so far I've not yet come with an answer which would require the use of college precalculus.
In a research facility in Taiwan a group of technicians built a new optical disk which stores information in a spiral engraved in its bottom face named "lecture side". Under the microscope it can be seen that the spiral begins in a region starting from $\textrm{2.6 cm}$ from its axis of rotation and it ends at $\textrm{5.7 cm}$ from the center of the disk. It can also be seen that individual turns of the spiral are $0.74\,\mu\textrm{m}$. Using this information calculate the length of the entire track.
So far I've only come with the idea of using the spiral of Archimedes, whose formula is given as follows:

$$r=a+b\phi$$

However, I'm not very familiar with the realm of polar coordinates or how can this equation be used to solve my problem.
To better illustrate the situation, however I've drawn this sketch to show how I'm understanding the problem.



I've included a cartesian grid, which well "may not be" in scale. But gives an idea of how I believe it is intended to be said.

I've really wanted to offer more into this such as an attempt into solving, but so far I've ran out of ideas.

However, I've come with the idea that the solution may be linked with finding how many turns are in the "readable sector" which is alluded in the problem.

To calculate this what I did was the following:

$\textrm{number of turns}= \left( 5.7 - 2.6 \right)\times 10^{-2}\textrm{m}\times \frac{\textrm{1 turn}}{0.74\times 10^{-6}\textrm{m}}$

By evaluating this short conversion factors I obtained

$\textrm{number of turns} = 41891.89189\,\textrm{turns or rad}$

And that's it. But from there I don't know how to relate this information with what would be needed to solve this riddle. The answer in the book states that the track's length is $1.09\times 10^{4}\,\textrm{m}$. But again. I don't know what can I do to get there.

Since I'm not very savvy with calculus and since this problem was obtained from a precalculus book, I aimed my question to that realm. Does it exist a method to make this computation relying in college algebra?. With this I don't intend to totally discard a calculus approach, but an answer with would really be very pedagogical for me is one which can show me the two methods, so I can compare which can be better according to my current knowledge so I can practice more in what I feel I'm lacking.

Well that's it. I do really hope somebody could help me with the two methods, one using algebra and another using calculus and more importantly how to use it.
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