Is this correct? wouldn't the radius of the circle it's revolving around be significantly smaller if you had it follow the edge of the larger circle but from the inside? I don't know much about math or physics, so I could be wrong, but I think it would be significantly less, closer to two, right?
The reason that this problem is tricky, and has a counterintuitive solution is that Circle A is rolling around Circle B, and so the 'radius; of the circular path it is following is the radius of Circle B + the radius of Circle A.
I'm no expert, but some quick math:
Circle B has a radius of 9, circumference then is 56.55
Circle A (1.3) has a radius of 3, circumference is 18.84
56.55 / 18.84 = 3. This suggests that you could "unwind" circle A (say it was made of pipecleaner), and you would need 3 circle As to fully ensconce Circle B
BUT that wasn't the question. The question states that Circle A is rolling AROUND circle B.
So the circumference of the circle we are now trying to 'ensconce' is Circle A Radius + Circle B Radius = 12, so the new circumference is 75.39, and divided by Circle A Circumference, we end up with 4, which makes sense, and matches the demonstration from the video.
HOWEVER, if we are 'rolling' around the inside of circle B, then I think you're right, the radius of the circular path Circle A will take is 9 -3 = 6, and the circumference of said circle is 38, so we it only will take two rolls. I think this is correct, i do not think it will be 4 rolls.
Think of it this way: when Circle A is rolling around the inside of Circle B, the way you're picturing it is circle A following the outside of Circle B, which is why I think intuitively it feels like the answer will still be 4 revolutions. BUT a better way to think of it is that Circle A is actually revolving around a new Circle, Circle C, which has a circumference of 38. Circle A is not revolving around Circle B, it's revolving around this new, smaller circle within circle b. Does that make sense?
I made a quick diagram to illustrate my point. I did it in a wire-framing tool that has snap to grid, so it's definitely not perfect:
The reason that this problem is tricky, and has a counterintuitive solution is that Circle A is rolling around Circle B, and so the 'radius; of the circular path it is following is the radius of Circle B + the radius of Circle A.
I'm no expert, but some quick math:
Circle B has a radius of 9, circumference then is 56.55 Circle A (1.3) has a radius of 3, circumference is 18.84
56.55 / 18.84 = 3. This suggests that you could "unwind" circle A (say it was made of pipecleaner), and you would need 3 circle As to fully ensconce Circle B
BUT that wasn't the question. The question states that Circle A is rolling AROUND circle B.
So the circumference of the circle we are now trying to 'ensconce' is Circle A Radius + Circle B Radius = 12, so the new circumference is 75.39, and divided by Circle A Circumference, we end up with 4, which makes sense, and matches the demonstration from the video.
HOWEVER, if we are 'rolling' around the inside of circle B, then I think you're right, the radius of the circular path Circle A will take is 9 -3 = 6, and the circumference of said circle is 38, so we it only will take two rolls. I think this is correct, i do not think it will be 4 rolls.
Think of it this way: when Circle A is rolling around the inside of Circle B, the way you're picturing it is circle A following the outside of Circle B, which is why I think intuitively it feels like the answer will still be 4 revolutions. BUT a better way to think of it is that Circle A is actually revolving around a new Circle, Circle C, which has a circumference of 38. Circle A is not revolving around Circle B, it's revolving around this new, smaller circle within circle b. Does that make sense?
I made a quick diagram to illustrate my point. I did it in a wire-framing tool that has snap to grid, so it's definitely not perfect:
https://i.imgur.com/dospt2w.png
You can play around with what I made here:
https://whimsical.com/r-DoXHwe2kUdgvSiAu2yuGpE