Prism and two balls HELP

Ana.stasia

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Sep 28, 2020
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A regular three-sided prism is described around a ball, and a ball is inscribed around it. Determine the ratio of the area of the balls.

This is my work. Where did I go wrong?
IMG_20201118_194532.jpg
IMG_20201118_194537.jpg
 

Dr.Peterson

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Nov 12, 2017
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A regular three-sided prism is described around a ball, and a ball is inscribed around it. Determine the ratio of the area of the balls.

This is my work. Where did I go wrong?
I would use the word "circumscribed" where you used both "described" and "inscribed", but I understand what you mean, I think.

I used a much simpler method and got the same answer you did, 5:1. So maybe you are actually right. (I could be wrong, too!)

What I did was to take the radius of the inner ball as r, and think about a cross-section of the figure through the center of the ball and a vertical side of the prism. This led to finding that the radius of the outer ball is \(\displaystyle r\sqrt{5}\), and therefore to your answer.

Make sure you looked at the answer to the right problem!

I can see a simple mistake they could have made to get an answer of 4:1. On the other hand, if the outer ball is not circumscribed about the prism (passing through all 6 of its vertices), but something else, then maybe we're solving the wrong problem.
 

Ana.stasia

New member
Joined
Sep 28, 2020
Messages
44
I would use the word "circumscribed" where you used both "described" and "inscribed", but I understand what you mean, I think.

I used a much simpler method and got the same answer you did, 5:1. So maybe you are actually right. (I could be wrong, too!)

What I did was to take the radius of the inner ball as r, and think about a cross-section of the figure through the center of the ball and a vertical side of the prism. This led to finding that the radius of the outer ball is \(\displaystyle r\sqrt{5}\), and therefore to your answer.

Make sure you looked at the answer to the right problem!

I can see a simple mistake they could have made to get an answer of 4:1. On the other hand, if the outer ball is not circumscribed about the prism (passing through all 6 of its vertices), but something else, then maybe we're solving the wrong problem.
I believe the outer ball is circumscribed about the prism because those are the only types of problems we have been learning to solve. Thank you for your input.
 
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