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run-of-the-mill math non-trick but possibly complicated question!?

casebro said:
Draw a circle of 1km in the road. It has a circumference of 3.14 km. Cut the circle in half at the road's center line. Slide one half along the path until the opposite ends met. You now have a sine wave with a length of 3.14km in 2km of road. Times 5,000 for how many 2km lengths of road. 15,700km.

Now, train the diver to spin the wheel from lock to lock each time he crosses the center line.

And there are different forms of sine waves, length vs amplitude, and 'slope' as it crosses center....
Click to expand...

Haha .. I actually encountered this when I was in my 5th grade and we were learning about sine for the first time. One friend of mine draw the sine curve using pair of compasses (? tool to draw circles) .. basically what you describe. Two half circles. I just couldn't explain to him it's completely wrong.

Sine curve simply isn't two half circles. The shape is completely different, and no scaling can change that. Half circles have the same curvature all along, suddenly switching direction when you go from one half circle to another. But sine curve doesn't do that. It changes the curvature fluently .. it is highest at it's peaks, and when it passes zero the curvature is zero. Derivation of sine function is cosine function (and vice versa).
 
Semantics.

I answered the OP with a formula and a logical process. The only one so far?

But a thought- is there a constant for sine waves (like yours, not mine, or both? ) that tells us length of path for amplitude and width? You said " The shape is completely different, and no scaling can change that. " hence my question.

eta3- and whre would it matter except in the OP puzzle?
 
Last edited:
Rewriting the question with a couple of the suggestions given in this thread just in case anyone wants to use this question sometime.


Two cars travel a perfectly straight road from Chicago, USA to Cairo, Egypt. (It's the future.) Each car is exactly 1 meter wide. The road is exactly 11 meters wide. The road is exactly 10,000 km long.

Car A travels a perfectly straight line down the center of the road the entire way. Car A therefore travels exactly 10,000 km.

Car B starts out on the side of the road (doesn't matter which side, but still technically on the road). The driver of Car B drifts from one side of the road to the other, every 1 km of road, the entire way.

How far exactly has Car B traveled?​



And I believe with these changes the answer is still 10,000.5 kilometers.

(Or, more specifically: 10,000.499987500624960940234169938 kilometers. :D)


Keeping in mind that we are assuming the car does an impossibly perfect zig-zag for simplicities sake. And that if we calculated for a realistic driving curve it would be a bit higher, around 10,000.617 kilometers (1234/2 meters extra. Thanks to Meridian in post #16.)
 
casebro said:
But a thought- is there a constant for sine waves (like yours, not mine, or both? ) that tells us length of path for amplitude and width?
Click to expand...

No, there is no closed-form solution using elementary functions.
 
Modified said:
No, there is no closed-form solution using elementary functions.
Click to expand...

gif.download


a,b = distance along highway
α = amplitude
ω = frequency (divided by 2π)
 
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