Length of a Curve Paradox. Help clarify?

Question

I was presented with the following problem:

In a fiber optic cable, the fiber optic follows a sine curve.  What is the ratio of the lengths of the cable to the length of the fiber within the cable?

My solution made me assume that the fiber optic inside the cable followed the path of the function y = sin x.  Logically, we can also assume that the fiber optic must be longer than the cable since it has a longer path to travel.  So, I focused my attention on a length of cable that was pi/2 units long, and I used the length of a curve formula to discover the length of the cable:

s = integral from 0 to pi/2 of (sqrt(1+cos^2 x)dx)
= integral from 0 to pi/2 of (sin x dx)
= - cos x evaluated from 0 to pi/2
= 1

So, the fiber optic cable is 1 unit long, and the cable itself is pi/2 units long.  The fiber optic is shorter than the cable itself?  Please help me discover the error in my calculation.

Anonymous · Accepted Answer

cos² x + sin² x = 1
sin² x = 1 - cos² x
sin x = sqrt(1 - cos² x)

So you messed up between the first and second lines of work, since you have a PLUS cos² x, rather than a minus.

mathman241 · Answer

the length of the cable is the length of the cable, the Embedding topology does not matter

2 pi R  == pi D  :: D

so 
L_cable  * Pi = L_fiber

ted s · Answer

thit is because you don't know your trig identities { careless??}....1 + cos ² x ╪ sin ² x...do the integral numerically on your calculator...you get about 1.9.....but note that you did not consider the amplitude of the function vs the cable diameter...{you assumed it to be 1 unit!!!}

Theta40 · Answer

1+cos^2 (x) is not sin^2(x)

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Length of a Curve Paradox. Help clarify?

4 Answers