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Math Challenge II-A Geometry, questions 9.27 & 9.30

 
 
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Math Challenge II-A Geometry, questions 9.27 & 9.30
by George Wen - Wednesday, November 27, 2019, 12:50 PM
 

How do you solve question 9.27 and 9.30? I can't imagine what the solids or shapes look like from the two problems. For question 9.27, what is S referring to? What does it look like? For question 9.30, how does the intersection of the sphere and frustum create a circle? 

 
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Re: Math Challenge II-A Geometry, questions 9.27 & 9.30
by Areteem Professor - Wednesday, November 27, 2019, 6:08 PM
 

For problem 27 here is a view of the cube and the tetrahedrons that are removed, and also a view of the created solid $\mathcal{S}$

and here's a view that lets us see the diagonal joining the two vertices from which the tetrahedra originate.


For problem 30, to see that the intersection of a sphere and frustrum create a circle: imagine you have a sphere and a circle with radius smaller than that of the sphere; then the sphere would not be able to go through the circle and would get stuck touching whole circle (you can literally try this with say a soccer ball for the sphere and a coffee mug for the circle). In the case of a cone the same thing would happen, as all the points of the sphere that are in the cone lie on the same plane parallel to the base of the cone.

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Re: Math Challenge II-A Geometry, questions 9.27 & 9.30
by George Wen - Thursday, November 28, 2019, 10:21 AM
 

Thank you so much! Could you explain what S means in question 9.27? I don't understand what the question is asking for. 

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Re: Math Challenge II-A Geometry, questions 9.27 & 9.30
by Areteem Professor - Monday, December 2, 2019, 11:12 AM
 

$\mathcal{S}$ is the name of the solid that was created. As the solid rests on different faces it's height (the distance from the bottom face to the top face) varies. The problem is asking to find all possible values of this height.