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Week 5 AMC10 Prep Course Homework

 
 
Picture of Duy Duong
Week 5 AMC10 Prep Course Homework
by Duy Duong - Thursday, 7 January 2021, 4:44 PM
 
A cube and a sphere share the same center. Each edge of the cube is half contained in the sphere. The ratio of the volume of the sphere to the cube can be written as Aπ:B for positive integer A and B with gcd(A,B)=1. What is A+B?


That is the question, I just need to know what it means by the edge of the cube is half contained in the sphere. Does it mean it is like inscribed or the sphere is bigger than it is? If you can, put a picture. That would help me a lot, thanks!

 
Picture of Charles Zhang
Re: Week 5 AMC10 Prep Course Homework
by Charles Zhang - Thursday, 7 January 2021, 6:34 PM
 

Sphere > Cube


Cube Edge:

__/____\__

  ^             ^

sphere intersections

Picture of Duy Duong
Re: Week 5 AMC10 Prep Course Homework
by Duy Duong - Friday, 8 January 2021, 4:23 PM
 

Thanks!

Picture of Daniel Zhu
Re: Week 5 AMC10 Prep Course Homework
by Daniel Zhu - Saturday, 9 January 2021, 2:41 PM
 

I also need a bit of help understanding this problem. Is the problem saying that the corners of the cube touch the sphere?

Picture of Charles Zhang
Re: Week 5 AMC10 Prep Course Homework
by Charles Zhang - Saturday, 9 January 2021, 3:04 PM
 
Imagine this: You have a sphere


Ok good


Now you ask your dad (or mom) to make some corners for you, that fit perfectly onto your sphere.


(Don't actually)


You glue them onto the sphere, so then it looks like a Pac-Man came alive and was tired of eating hosts, so he decided to devour about half of a cube.


(Make sure that there is a bump on every "face" of the "cube" you just made, and the cross sections look like a circle with edges sticking out at 90 degree angles)


That's what this problem is describing.

Picture of Charles Zhang
Re: Week 5 AMC10 Prep Course Homework
by Charles Zhang - Saturday, 9 January 2021, 3:01 PM
 

And yes this is along the lines of what I actually picture in my head when I did this.

Picture of Daniel Zhu
Re: Week 5 AMC10 Prep Course Homework
by Daniel Zhu - Sunday, 10 January 2021, 2:57 PM
 

lol thanks!