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A glass is shaped like a right circular cylinder with a half sphere at

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A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 13 Apr 2011, 11:16
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A glass is shaped like a right circular cylinder with a half sphere at the bottom. The glass is 7cm deep and has a diameter of 6cm, measured on the inside. If the glass is filled to the rim with apple cider, how much cider is in the glass? The formula for the volume a sphere is \(\frac{4}{3}*\pi*r^3\).

A. 18\(\pi\)
B. 36\(\pi\)
C. 54\(\pi\)
D. 81\(\pi\)
E. 108\(\pi\)


I'm confusing about the part "If the glass is filled to the rim with apple cider..." , how do we understand this problem?
Attachment:
cup.png
cup.png [ 1.2 KiB | Viewed 851 times ]
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 13 Apr 2011, 12:35
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yvonne0923 wrote:
A glass is shaped like a right circular cylinder with a half sphere at the bottom. The glass is 7cm deep and has a diameter of 6cm, measured on the inside. If the glass is filled to the rim with apple cider, how much cider is in the glass? The formula for the volume a sphere is \(3/4*Pi*r^3\).

A. 18Pi
B. 36Pi
C. 54Pi
D. 81Pi
E. 108Pi

I'm confusing about the part "If the glass is filled to the rim with apple cider..." , how do we understand this problem?


I don't understand the meaning of this expression. Assuming it's trying to say "filled up to the brim" (fully filled).

And the volume of the sphere is:
\(\frac{4}{3}*\pi*r^3\)

The glass looks something like the one in the attached figure.

Attachment:
glass_with_cider.PNG
glass_with_cider.PNG [ 9.17 KiB | Viewed 14409 times ]


All we need to do is find the volume of the glass, which is volume of the hemispherical base+volume of the cylindrical top.

Volume of hemisphere = 1/2*(Volume of sphere) \(= \frac{1}{2}*\frac{4}{3}*\pi*r^3\), where r=radius of hemisphere

Volume of cylinder \(= \pi*r^2*h\), where r=radius of cylinder & h=height of cylinder

From the figure, we can see that Radius of cylinder=Radius of hemisphere
Also, the total depth of the glass= Radius of hemisphere+height of cylinder


glass has a diameter of 6cm
\(r=3 cm\)

glass is 7cm deep
\(r+h=7\)
\(3+h=7\)
\(h=4cm\)

Total Volume of the glass= Volume of hemisphere+Volume of cylinder

\(V=\frac{1}{2}*\frac{4}{3}*\pi*r^3+\pi*r^2*h\)
\(V=\frac{2}{3}*\pi*3^3+\pi*3^2*4\)
\(V=18\pi+36\pi=54\pi\)

Ans: "C"
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 13 Apr 2011, 15:35
fluke wrote:
yvonne0923 wrote:
A glass is shaped like a right circular cylinder with a half sphere at the bottom. The glass is 7cm deep and has a diameter of 6cm, measured on the inside. If the glass is filled to the rim with apple cider, how much cider is in the glass? The formula for the volume a sphere is \(3/4*Pi*r^3\).

A. 18Pi
B. 36Pi
C. 54Pi
D. 81Pi
E. 108Pi

I'm confusing about the part "If the glass is filled to the rim with apple cider..." , how do we understand this problem?


I don't understand the meaning of this expression. Assuming it's trying to say "filled up to the brim" (fully filled).

And the volume of the sphere is:
\(\frac{4}{3}*\pi*r^3\)

The glass looks something like the one in the attached figure.

Attachment:
The attachment glass_with_cider.PNG is no longer available


All we need to do is find the volume of the glass, which is volume of the hemispherical base+volume of the cylindrical top.

Volume of hemisphere = 1/2*(Volume of sphere) \(= \frac{1}{2}*\frac{4}{3}*\pi*r^3\), where r=radius of hemisphere

Volume of cylinder \(= \pi*r^2*h\), where r=radius of cylinder & h=height of cylinder

From the figure, we can see that Radius of cylinder=Radius of hemisphere
Also, the total depth of the glass= Radius of hemisphere+height of cylinder


glass has a diameter of 6cm
\(r=3 cm\)

glass is 7cm deep
\(r+h=7\)
\(3+h=7\)
\(h=4cm\)

Total Volume of the glass= Volume of hemisphere+Volume of cylinder

\(V=\frac{1}{2}*\frac{4}{3}*\pi*r^3+\pi*r^2*h\)
\(V=\frac{2}{3}*\pi*3^3+\pi*3^2*4\)
\(V=18\pi+36\pi=54\pi\)

Ans: "C"





I understand how to solve the problem if there is no image attached, but the image shown on the book is like a shape of glass wine below.

Attachment:
Glass_Cider.jpg
Glass_Cider.jpg [ 1.87 KiB | Viewed 14390 times ]


So if in this case, I believe that the cider is impossible to reach the holder of the glass, and I supposed to substract the height of the holder from the total height of glass of 7cm. However, how can we know the height of the holder? Also, in this case, the information of sphere volume is not really necessary here according to this image.
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 13 Apr 2011, 19:37
yvonne0923 wrote:


I understand how to solve the problem if there is no image attached, but the image shown on the book is like a shape of glass wine below.

Attachment:
Glass_Cider.jpg


So if in this case, I believe that the cider is impossible to reach the holder of the glass, and I supposed to substract the height of the holder from the total height of glass of 7cm. However, how can we know the height of the holder? Also, in this case, the information of sphere volume is not really necessary here according to this image.


Actually, the image has been given to help you understand the question better, not confuse you. The solution is as given by fluke. Look at the figure drawn by fluke. It is the top part of the glass and that is what matters. The entire information given is for the top part only. The stem of the glass is there to just hold it and doesn't account for anything (It has been drawn so that people don't worry about how the glass will 'stand' with a round bottom).

Read the question again. Let me quote:
'The glass is 7cm deep and has a diameter of 6cm, measured on the inside. If the glass is filled to the rim with apple cider, how much cider is in the glass?'

The glass i.e. the part that will hold the liquid, is 7 cm deep so from top of cylinder to lowest point of hemisphere i.e. just the part shown by fluke is 7 cm.
The rim is the outer edge or the brink of a circular object. When we say it is filled to the rim, it means it is filled to the top i.e. the part shown by fluke in the diagram is completely filled. Just think of an actual wine glass filled to the top. Now focus on the part that holds the wine - that is the part whose dimensions are given. Forget the stem and the base. They are drawn just to satisfy you that we are talking about an actual glass.
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 13 Apr 2011, 19:55
Total Vol of wine = Vol of wine in Cylinder part + Vol of wine in Hemishphere

= pi*3^2* (7-3) + 2/3 * pi * 3^3

= 36pi + 18pi

= 54pi

Answer - C
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 14 Apr 2011, 10:42
fluke wrote:
yvonne0923 wrote:
A glass is shaped like a right circular cylinder with a half sphere at the bottom. The glass is 7cm deep and has a diameter of 6cm, measured on the inside. If the glass is filled to the rim with apple cider, how much cider is in the glass? The formula for the volume a sphere is \(3/4*Pi*r^3\).

A. 18Pi
B. 36Pi
C. 54Pi
D. 81Pi
E. 108Pi

I'm confusing about the part "If the glass is filled to the rim with apple cider..." , how do we understand this problem?


I don't understand the meaning of this expression. Assuming it's trying to say "filled up to the brim" (fully filled).

And the volume of the sphere is:
\(\frac{4}{3}*\pi*r^3\)

The glass looks something like the one in the attached figure.

Attachment:
glass_with_cider.PNG


All we need to do is find the volume of the glass, which is volume of the hemispherical base+volume of the cylindrical top.

Volume of hemisphere = 1/2*(Volume of sphere) \(= \frac{1}{2}*\frac{4}{3}*\pi*r^3\), where r=radius of hemisphere

Volume of cylinder \(= \pi*r^2*h\), where r=radius of cylinder & h=height of cylinder

From the figure, we can see that Radius of cylinder=Radius of hemisphere
Also, the total depth of the glass= Radius of hemisphere+height of cylinder


glass has a diameter of 6cm
\(r=3 cm\)

glass is 7cm deep
\(r+h=7\)
\(3+h=7\)
\(h=4cm\)

Total Volume of the glass= Volume of hemisphere+Volume of cylinder

\(V=\frac{1}{2}*\frac{4}{3}*\pi*r^3+\pi*r^2*h\)
\(V=\frac{2}{3}*\pi*3^3+\pi*3^2*4\)
\(V=18\pi+36\pi=54\pi\)

Ans: "C"


Dear Fluke
as the figure was not attached how can you say that the sphere is not placed upside down
i arrived at 45Pi as i was deducting the quantity (as i thought it to be upside down)
I think it was only given options from which i could thought of adding instead of deducting the volume of sphere from the total volume of cylinder :(

please advice
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 14 Apr 2011, 12:14
fluke wrote:
Warlock007 wrote:
Dear Fluke
as the figure was not attached how can you say that the sphere is not placed upside down
Intuition and good luck!!! :) I see that you made a genuine point here. I should have been more cautious.

i arrived at 45Pi as i was deducting the quantity (as i thought it to be upside down)
How did you arrive at \(45\pi\)? Even if the hemisphere is upside down at the bottom, the result should be \(72\pi\). Please explain. On a second thought, now that we know the hemisphere is not upside down, do we really need to do this? Up to you.

I think it was only given options from which i could thought of adding instead of deducting the volume of sphere from the total volume of cylinder :(

please advice



I have taken the hemisphere volume to be \(18\pi\) only but in case of cylinder i have taken h=7
I dont know how come you deducted the radius from the given depth 7 cm

in my approach total volume is = \(63\pi\) + \(18\pi\) =\(81\pi\)
in your approach total volume is = \(63\pi\) - \(18\pi\) =\(45\pi\)

and sorry my pick was D \(81\pi\)

please correct my approach
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 14 Apr 2011, 12:35
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Warlock007 wrote:
fluke wrote:
Warlock007 wrote:
Dear Fluke
as the figure was not attached how can you say that the sphere is not placed upside down
Intuition and good luck!!! :) I see that you made a genuine point here. I should have been more cautious.

i arrived at 45Pi as i was deducting the quantity (as i thought it to be upside down)
How did you arrive at \(45\pi\)? Even if the hemisphere is upside down at the bottom, the result should be \(72\pi\). Please explain. On a second thought, now that we know the hemisphere is not upside down, do we really need to do this? Up to you.

I think it was only given options from which i could thought of adding instead of deducting the volume of sphere from the total volume of cylinder :(

please advice



I have taken the hemisphere volume to be \(18\pi\) only but in case of cylinder i have taken h=7
I dont know how come you deducted the radius from the given depth 7 cm

Depth of the glass = 7cm
If we go by the original question and original figure of the question,
7cm=Radius of hemisphere+height of cylinder
And it is given that Radius of hemisphere=3cm
7cm=3cm+height of cylinder
height of cylinder=7cm-3cm=4cm

Please see the attached image:
Red line+Green line=Depth of the glass=7cm
Red line=Radius of hemisphere= 3cm
Green line=Height of cylinder= 4cm

So, Volume of cylinder= \(\pi*r^2*h=\pi*3^2*4=36\pi\)

Attachment:
glass_with_cider.PNG
glass_with_cider.PNG [ 9.54 KiB | Viewed 14279 times ]



in my approach total volume is = \(63\pi\) + \(18\pi\) =\(81\pi\)
in your approach total volume is = \(63\pi\) - \(18\pi\) =\(45\pi\)

and sorry my pick was D \(81\pi\)

please correct my approach
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 14 Apr 2011, 12:41
fluke wrote:
Dear Fluke
as the figure was not attached how can you say that the sphere is not placed upside down
Intuition and good luck!!! :) I see that you made a genuine point here. I should have been more cautious.

i arrived at 45Pi as i was deducting the quantity (as i thought it to be upside down)
How did you arrive at \(45\pi\)? Even if the hemisphere is upside down at the bottom, the result should be \(72\pi\). Please explain. On a second thought, now that we know the hemisphere is not upside down, do we really need to do this? Up to you.

I think it was only given options from which i could thought of adding instead of deducting the volume of sphere from the total volume of cylinder :(

please advice
[/quote]


I have taken the hemisphere volume to be \(18\pi\) only but in case of cylinder i have taken h=7
I dont know how come you deducted the radius from the given depth 7 cm

Depth of the glass = 7cm
If we go by the original question and original figure of the question,
7cm=Radius of hemisphere+height of cylinder
And it is given that Radius of hemisphere=3cm
7cm=3cm+height of cylinder
height of cylinder=7cm-3cm=4cm

Please see the attached image:
Red line+Green line=Depth of the glass=7cm
Red line=Radius of hemisphere= 3cm
Green line=Height of cylinder= 4cm

So, Volume of cylinder= \(\pi*r^2*h=\pi*3^2*4=36\pi\)

Attachment:
glass_with_cider.PNG



in my approach total volume is = \(63\pi\) + \(18\pi\) =\(81\pi\)
in your approach total volume is = \(63\pi\) - \(18\pi\) =\(45\pi\)

and sorry my pick was D \(81\pi\)

please correct my approach[/quote][/quote]

Dear Fluke thats what confused me should i include or not the radius

actually coincidently i found \(81\pi\) in the answers too
I hope this kind of disaster would not happen in real GMAT :( :(

kudos to you :-D :-D
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post Updated on: 17 Feb 2016, 10:50
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Nevernevergiveup wrote:
Attachment:
cup.png

A glass is shaped like a right circular cylinder with a half sphere at the bottom. The glass is 7cm deep and has a diameter of 6cm, measured on the inside. If the glass is filled to the rim with apple cider, how much cider is in the glass? The formula for the volume a sphere is \(\frac{4}{3}*\pi*r^3\).

A. 18\(\pi\)
B. 36\(\pi\)
C. 54\(\pi\)
D. 81\(\pi\)
E. 108\(\pi\)


The glass volume is composed of 2 parts: hemispherical portion + cylindrical portion.

For the hemispherical portion, diameter = 6 cm ---> radius = r= 3 ---> volume of the hemisphere = volume of sphere /2 = \(0.5*\frac{4}{3}*\pi*r^3 = 18\pi\)

For the cylindrical part, heigh=h=7-3 = 4 and radius =3 ---> volume of the cylindrical portion \(= \pi*r^2*h = 36\pi\)

Thus the total volume =\(( 18+36 )\pi = 54\pi\)

C is thus the correct answer.

Hope this helps.

Originally posted by ENGRTOMBA2018 on 17 Feb 2016, 10:38.
Last edited by ENGRTOMBA2018 on 17 Feb 2016, 10:50, edited 1 time in total.
Corrected the typo.
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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New post 06 Jun 2018, 11:47
yvonne0923 wrote:
A glass is shaped like a right circular cylinder with a half sphere at the bottom. The glass is 7cm deep and has a diameter of 6cm, measured on the inside. If the glass is filled to the rim with apple cider, how much cider is in the glass? The formula for the volume a sphere is \(\frac{4}{3}*\pi*r^3\).

A. 18Pi
B. 36Pi
C. 54Pi
D. 81Pi
E. 108Pi

I'm confusing about the part "If the glass is filled to the rim with apple cider..." , how do we understand this problem?


Voulume of Cidar in the Glass = Vol. of Cylinder - Vol. of Semisphere + Area of the Base of Cylinder
= \(πr^2h - 2/3πr^3 + πr^2\)
= \(54π\)
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Re: A glass is shaped like a right circular cylinder with a half sphere at  [#permalink]

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