Math challenge!

Ok, you have two circles such as these: http://www.cut-the-knot.org/Outline/...gCircles.shtml

The bigger circle has the equation x^2+y^2=16. The smaller circle has a radius of 1. 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?

Hey MJ, whats your IQ.

Hey MJ, whats your IQ.

Lol, that's not relevant. More importantly, I don't know. It's not that complicated of a problem. If nobody gets it in a couple of days, I'll post a hint.

It's not that complicated of a problem.

WTF. I don't even know what those small upper-type arrows are.

WTF. I don't even know what those small upper-type arrows are.

"x squared"

^ = exponent

Now that you've blessed me. Lemme give it a try.

Edit: Nevermind I came across this shit "||" and got stumped.

I quit.

...the equal sign?

Yeah the one that goes downwards though.

The equal sign has no parallel.

"| |" is the modulus of something - the modulus is the positive values.

So:

|3| = 3
|-3| = 3

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The site won't load for me, so I'm doing this without a nice diagram... glee.

I'm probably thinking far too simply, but would the area just be that of the bigger circle?

No. I'll try and drop up a diagram of how it should look.

The red dot is a point on the r=1 circle. The r=1 circle needs to fully traverse the circumference of the r=4 circle.

Ah right.

I'll give it a shot after my pizza.

Geometry hurts me.

Hint: The parametric equations of the path created by the point on circle r=1 are x(t)=5cos(t)+cos(5t), y(t)=5sin(t)+sin(5t), where 0<=t<=2*pi

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...

I definently thought that was the congruent symbol hahaha... but i hate math...

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...