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# A right circular cone is inscribed in a hemisphere so that

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A right circular cone is inscribed in a hemisphere so that [#permalink]

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03 Apr 2012, 16:49
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A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere?

A. $$\sqrt{3} : 1$$

B. $$1 : 1$$

C. $$\frac{1}{2} : 1$$

D. $$\sqrt{2} : 1$$

E. $$2 : 1$$

I've never heard of a "hemisphere". I know that the cone forms a 30-60-90 right triangle and therefore the height of the cone is /3 but because I don't know what to do with the hemisphere, I'm stuck.

Thanks,
Rich
[Reveal] Spoiler: OA

Last edited by Bunuel on 04 Apr 2012, 00:23, edited 1 time in total.
Edited the question

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03 Apr 2012, 17:41
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A hemisphere is half of a sphere.

A right cone does not necessarily form a 30-60-90 triangle. In this case, the cone's is formed from an a isosceles right triangle rotated about the center - so the radius of the cone's base (also the hemisphere's base) = the height of the cone.

1:1 ratio - B

I noticed some "gimme" geometry problem on my exams - learn this stuff well & you can shave minutes off the time spent on geometry questions.

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03 Apr 2012, 20:18
nsspaz151 wrote:
A hemisphere is half of a sphere.

A right cone does not necessarily form a 30-60-90 triangle. In this case, the cone's is formed from an a isosceles right triangle rotated about the center - so the radius of the cone's base (also the hemisphere's base) = the height of the cone.

1:1 ratio - B

I noticed some "gimme" geometry problem on my exams - learn this stuff well & you can shave minutes off the time spent on geometry questions.

I take back what I said regarding the 30-60-90 triangle, that was a silly assumption on my part.

Thanks for the explanation regarding the hemisphere, it's so simple!

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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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04 Apr 2012, 00:48
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NvrEvrGvUp wrote:
A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere?

A. $$\sqrt{3} : 1$$

B. $$1 : 1$$

C. $$\frac{1}{2} : 1$$

D. $$\sqrt{2} : 1$$

E. $$2 : 1$$

I've never heard of a "hemisphere". I know that the cone forms a 30-60-90 right triangle and therefore the height of the cone is /3 but because I don't know what to do with the hemisphere, I'm stuck.

Thanks,
Rich

As mentioned above hemisphere is just a half of a sphere. Now, since the cone is a right circular cone, the vertex of the cone must touch the surface of the hemisphere directly above the center of the base, which makes the height of the cone also the radius of the hemisphere.

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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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23 Nov 2012, 02:10
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Not sure if anyone else was scratching their heads wondering why the height couldn't be less than the radius, but just in case... it has to do with the word "inscribed": to draw within a figure so as to touch in as many places as possible <a regular polygon inscribed in a circle>

lol.

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04 Jan 2013, 09:23
nsspaz151 wrote:
A hemisphere is half of a sphere.

A right cone does not necessarily form a 30-60-90 triangle. In this case, the cone's is formed from an a isosceles right triangle rotated about the center - so the radius of the cone's base (also the hemisphere's base) = the height of the cone.

1:1 ratio - B

I noticed some "gimme" geometry problem on my exams - learn this stuff well & you can shave minutes off the time spent on geometry questions.

"gimme" geometry problems? might be because I'm new on the forum, but can you tell me what are you referring to?

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04 Jan 2013, 10:12
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sianissimo wrote:
nsspaz151 wrote:
A hemisphere is half of a sphere.

A right cone does not necessarily form a 30-60-90 triangle. In this case, the cone's is formed from an a isosceles right triangle rotated about the center - so the radius of the cone's base (also the hemisphere's base) = the height of the cone.

1:1 ratio - B

I noticed some "gimme" geometry problem on my exams - learn this stuff well & you can shave minutes off the time spent on geometry questions.

"gimme" geometry problems? might be because I'm new on the forum, but can you tell me what are you referring to?

"Gimme" is slang for quick/easy. If you know geometry well, you can avoid doing any calculation on some problems.

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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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08 Jun 2013, 21:12
I know there are solutions to this question posted, however I have a follow-up clarifying question.

A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere?

My question is, if you have a cone with a radius of 5 and a height of 1, how could the ratio be 1:1? You could have a very wide and short hemisphere. Am I missing something with regards to the cone being designated as a "right circular cone"? Can someone please explain?

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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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08 Jun 2013, 22:36
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It's the combination of "right circular cone" and "inscribed". The cone shares a base with the hemisphere and it's tip touches the top of the hemisphere. If you had a right circular cone of a height to radius ratio of anything other than 1:1 you would not be able to inscribe it in a hemisphere. Essentially, what you've described is impossible for this particular problem.

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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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08 Jun 2013, 22:39
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Sorry, forgot to add that you cannot have a hemisphere of the type you've described. It wouldn't be a hemisphere. A sphere's radius is the same throughout. A hemisphere is half of a sphere (so it's height must equal it's radius.

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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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01 Aug 2014, 04:01
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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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29 May 2017, 23:56
essarr wrote:
Not sure if anyone else was scratching their heads wondering why the height couldn't be less than the radius, but just in case... it has to do with the word "inscribed": to draw within a figure so as to touch in as many places as possible <a regular polygon inscribed in a circle>

lol.

Exactly the query I had, thanks for giving that solution !

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A right circular cone is inscribed in a hemisphere so that [#permalink]

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07 Jul 2017, 06:06
Bunuel

It makes sense that the height of the cone will be equal to the radius of the hemisphere.

However, when I tried equating the volumes for both the cone and hemisphere, the solution I am getting is 2 : 1

1\3 x pie x r^2 x h = 1/2 x 4/3 x pie x r^3

h = 2r

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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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07 Jul 2017, 06:23
kotharishakti wrote:
Bunuel

It makes sense that the height of the cone will be equal to the radius of the hemisphere.

However, when I tried equating the volumes for both the cone and hemisphere, the solution I am getting is 2 : 1

1\3 x pie x r^2 x h = 1/2 x 4/3 x pie x r^3

h = 2r

Why are you equating volumes? The volume of a cone is obviously less than the volume of the hemisphere it is inscribed in.
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Re: A right circular cone is inscribed in a hemisphere so that [#permalink]

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07 Jul 2017, 07:20
Bunuel wrote:
kotharishakti wrote:
Bunuel

It makes sense that the height of the cone will be equal to the radius of the hemisphere.

However, when I tried equating the volumes for both the cone and hemisphere, the solution I am getting is 2 : 1

1\3 x pie x r^2 x h = 1/2 x 4/3 x pie x r^3

h = 2r

Why are you equating volumes? The volume of a cone is obviously less than the volume of the hemisphere it is inscribed in.

Oh yes!!
That's Correct Bunuel.

I am doing it wrong! Such a silly one.

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Re: A right circular cone is inscribed in a hemisphere so that   [#permalink] 07 Jul 2017, 07:20
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