Figure 1 is formed by a rectangular box measuring 38cm by 15cm and 2 identical white balls...

stapel said:
What are your thoughts? What have you tried? Where are you getting stuck?

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I have tried to understand the problem, but I am getting stuck in the following areas:

  • I am not sure how to calculate the distance that ball A has rolled along figure 1 when it returns back to its original position.
  • I am not sure what the significance of the fact that ball A is the same size as the two identical white balls is.
  • I am not sure what the significance of the fact that the two identical white balls are fixed to the box at a point is.
 
ydubrovensky4 said:
I am not sure how to calculate the distance that ball A has rolled along figure 1 when it returns back to its original position.
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Think about the path the center of the ball will follow, and add up the lengths of the segments and arcs that make up that path. Here is a picture of part of that path, in red:

1695598817308.png​

ydubrovensky4 said:
I am not sure what the significance of the fact that ball A is the same size as the two identical white balls is.
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Without knowing that, you couldn't find the path. Look at the picture above.

ydubrovensky4 said:
I am not sure what the significance of the fact that the two identical white balls are fixed to the box at a point is.
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They determine part of the path.

But there's one really big thing you are misunderstanding, as shown by your drawing:

1695598932678.png​

They tell you that the ball rolls anti-clockwise around the entire figure, not just back and forth along the top. So the path is more like this:
1695599128495.png​

Does that help you visualize it?
 
Steven G said:
Dr Peterson,
Does the ball start at point D or 3.5 cm from point D?
Thank you,
Steven
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It doesn't say where it starts; you can take your choice. It doesn't matter, since it returns to wherever it starts.

I would probably take it as starting at D for simplicity, if I paid any attention to that at all. I didn't bother showing that in the last picture. Is this clearer?
1695610772489.png​
 
Dr.Peterson said:
It doesn't say where it starts; you can take your choice. It doesn't matter, since it returns to wherever it starts.

I would probably take it as starting at D for simplicity, if I paid any attention to that at all. I didn't bother showing that in the last picture. Is this clearer?
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OK, I see what you're saying. I too realized that where it starts didn't matter. I was just curious why you chose to start at D
 
Steven G said:
OK, I see what you're saying. I too realized that where it starts didn't matter. I was just curious why you chose to start at D
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Um ... because that's (near) where the ball is in the picture I used?

I didn't actually say to start either at D, or where the ball is shown, in the first place. That's your assumption. All I did was to modify a picture to show the meaning of the problem, without saying anything about the calculation to do. In fact, now that I think about it, I wouldn't actually start anywhere. I'd just add 12 distances. (As I said, "if I paid any attention to that at all.")
 
Dr.Peterson said:
Without knowing that, you couldn't find the path. Look at the picture above.

They determine part of the path.

But there's one really big thing you are misunderstanding, as shown by your drawing:

1695598932678.png

They tell you that the ball rolls anti-clockwise around the entire figure, not just back and forth along the top. So the path is more like this:
1695599128495.png

Does that help you visualize it?
Click to expand...
On the 1/2O, I got 1/2 * 3.14 * 7
 
The diameter of the rolling ball is equal to the diameter of the fixed balls. When the ball rolls around each ball, it covers half of its circumference. The total distance of the ball when rolling around each ball is its circumference. The circumference of a rectangle is equal to the square of the diameter of the rectangle. The square of the rectangle is equal to 7 (pi + diameter = 3.14). Since the rectangle has a circumference of 31 cm at the long edges of the rectangle and 15 cm at the short edges of the rectangle, the total circumference of the ball once is 113.98 cm.
 
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