Math challenge!

Here's a better picture:



It's like the one on the right.

Ah, that makes more sense.

The question confused me somewhat:

If we were to have the smaller circle go around the bigger one (as though the bigger one were a surface and the smaller one were a tire) and we kept track of a single point on the edge of the smaller tire and drew a dot on every point it touched on the plane, once the smaller tire had made a complete revolution around the bigger one, what would the total enclosed area be that the point on the smaller tire made?

To me that meant a dot everytime it touched the other circle, and thus the area enclosed by those dots - the area of the bigger circle.

I've got a vague idea of how I might solve it, so I might get it done during my free's tomorrow. Or give it to one of my Uber Maths Genius friends and see what they make of it.

EDIT: I wonder if I can somehow make it a graph and use integration to find the area of each curve over the circle, and then add in the area of the circle...

EDIT: I wonder if I can somehow make it a graph and use integration to find the area of each curve over the circle, and then add in the area of the circle...

You're on the right track.

Find the equation of the line the smaller circle with the dot makes - shown as a red line on the animated diagram.

Using my integration thing - I know that the small circle will turn 360 degrees 16 times before it goes around the whole of the bigger circle, so if I put the red lines on to a graph (all 16 of them), with the corresponding portion of the bigger circle circumference (I think it will be 2 x pi units long), and then integrate to find the area below the red line, and then to find the area below the circle line, and take the circle area from the red line area (either do this sixteen times, or once and multiply the answer by sixteen), and then add in the full area of the circle.

The only work I'll need to do is work out the equations of each line... which I can probably do if I can somehow draw a accurate graph.

Find the equation of the line the smaller circle with the dot makes - shown as a red line on the animated diagram. Using my integration thing - I know that the small circle will turn 360 degrees 16 times

8 times...

Try a double integral with polar coordinates.

Eight, sixteen, what's the difference?

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Not been taught double integration yet. In fact, I don't think I will be taught it unless I do Further Maths, or go on to Degree Level Maths.

Either way, it's not on the A-Level Mathematics syllabus.

At first glance every time I see this thread I read "Meth Challenge!"