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The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 70% (02:36) correct 30% (02:34) wrong based on 519 sessions

### HideShow timer Statistics The volume of a sphere with radius r is $$\frac {4}{3} \pi r^3$$. A solid sphere of radius r that is made of a certain material weighs 40 pounds. The figure above shows half of a spherical shell made of the same material. What is the weight, in pounds, of the entire spherical shell if the outer radius is 5r and the inner radius is 2r?

A. 120
B. 360
C. 840
D. 1,080
E. 4,680

Attachment: Untitled.png [ 2.03 KiB | Viewed 50663 times ]
Attachment: GmatPrepImage.png [ 19.81 KiB | Viewed 39998 times ]

Originally posted by srmessi on 16 May 2016, 23:14.
Last edited by Bunuel on 17 May 2016, 00:38, edited 2 times in total.
Renamed the topic and edited the question.
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The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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srmessi wrote:
Can any1 help me with This

Hi,

the V = $$\frac{4}{3}pi*r^3$$ and this volume weighs 40 pounds...

If it is a shell, you can find V of solid part by subtracting inner empty sphere from total..
V of entire sphere $$\frac{4}{3} * pi *(5r)^3$$and V of inner empty space = $$\frac{4}{3} * pi *(2r)^3$$..

so V of solid =$$\frac{4}{3} * pi *(5r)^3 - \frac{4}{3}* pi *(2r)^3........................ = \frac{4}{3}*pi*r^3*(125-8)....................... = \frac{4}{3}pi*r^3*117$$

NOW $$\frac{4}{3}pi*r^3$$ weighs 40 pounds,
so $$\frac{4}{3}pi*r^3*117$$ will weigh $$40*117 = 4680$$...

E
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Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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A very tricky question
Weight of the sphere with radius r is =40
Now the radius of the outer shell =5r if we consider it as sphere then its weight would be 125 times 40=5000
Now the sphere with inner radius 2r will have 8 times 40 as weight =320
Subtract the weight ans we get the weight of the shell =5000-320=4680
##### General Discussion
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The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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srmessi wrote: The volume of a sphere with radius r is $$\frac {4}{3} \pi r^3$$. A solid sphere of radius r that is made of a certain material weighs 40 pounds. The figure above shows half of a spherical shell made of the same material. What is the weight, in pounds, of the entire spherical shell if the outer radius is 5r and the inner radius is 2r?

A. 120
B. 360
C. 840
D. 1,080
E. 4,680

Attachment:
Untitled.png
Attachment:
GmatPrepImage.png

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Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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srmessi wrote: The volume of a sphere with radius r is $$\frac {4}{3} \pi r^3$$. A solid sphere of radius r that is made of a certain material weighs 40 pounds. The figure above shows half of a spherical shell made of the same material. What is the weight, in pounds, of the entire spherical shellif the outer radius is 5r and the inner radius is 2r?

A. 120
B. 360
C. 840
D. 1,080
E. 4,680

Attachment:
The attachment Untitled.png is no longer available
Attachment:
The attachment GmatPrepImage.png is no longer available

$$\frac {4}{3} \pi r^3$$ = $$40$$

$$\frac{22}{7} r^3$$ = $$30$$

$$\frac{11}{7} r^3$$ = $$15$$

$$r^3$$ = $$15*7/11$$

Now think about the semi circle

Attachment: Circle.PNG [ 13.33 KiB | Viewed 39672 times ]

So, Volume will be = $$\frac{2}{3}*\frac{22}{7}*( 5r^3 - 2r^3)$$

Or Volume = $$\frac{2}{3}*\frac{22}{7}*( 117r^3)$$

We know r^3 = $$15*7/11$$ ; so substitute it ....

Volume = $$\frac{4}{3}*\frac{22}{7}*( 117*15*\frac{7}{11})$$ ( Check the highlighted part in the question we require the volume of the complete sphere )

Volume = 4,680

You may ask the question why the question mentions

Quote:
The figure above shows half of a spherical shell

It's given only to show that the thickness of the spherical ball (which is hollow from inside )

Hence I completely endorse answer option (E)
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Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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chetan2u wrote:
srmessi wrote:
Can any1 help me with This

Hi,

the V = $$\frac{4}{3}pi*r^3$$ and this volume weighs 40 pounds...

If it is a shell, you can find V of solid part by subtracting inner empty sphere from total..
V of entire sphere $$\frac{4}{3} * pi *(5r)^3$$and V of inner empty space = $$\frac{4}{3} * pi *(2r)^3$$..

so V of solid =$$\frac{4}{3} * pi *(5r)^3 - \frac{4}{3}* pi *(2r)^3........................ = \frac{4}{3}*pi*r^3*(125-8)....................... = \frac{4}{3}pi*r^3*117$$

NOW $$\frac{4}{3}pi*r^3$$ weighs 40 pounds,
so $$\frac{4}{3}pi*r^3*117$$ will weigh $$40*117 = 4680$$...

E

The way you explained was really good, thanks a lot.
One small question, how did you consider v=w?
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Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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Buttercup3 wrote:
chetan2u wrote:
srmessi wrote:
Can any1 help me with This

Hi,

the V = $$\frac{4}{3}pi*r^3$$ and this volume weighs 40 pounds...

If it is a shell, you can find V of solid part by subtracting inner empty sphere from total..
V of entire sphere $$\frac{4}{3} * pi *(5r)^3$$and V of inner empty space = $$\frac{4}{3} * pi *(2r)^3$$..

so V of solid =$$\frac{4}{3} * pi *(5r)^3 - \frac{4}{3}* pi *(2r)^3........................ = \frac{4}{3}*pi*r^3*(125-8)....................... = \frac{4}{3}pi*r^3*117$$

NOW $$\frac{4}{3}pi*r^3$$ weighs 40 pounds,
so $$\frac{4}{3}pi*r^3*117$$ will weigh $$40*117 = 4680$$...

E

The way you explained was really good, thanks a lot.
One small question, how did you consider v=w?

Hi..

It is not that the volume and weight is being equalled.
We are finding a relationship between the two as the weight will depend on the volume
Say you have a cake with a weight of 20 pounds and you cut 10 piece. What will be the weight of each piece. Ofcourse 20/10..
Here we are talking of volume and not pieces but finally that entire cake has a certain volume and by making 10 EQUAL parts, we are cutting the volume in 10 parts.
Otherwise volume*density= weight.
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The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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srmessi wrote: The volume of a sphere with radius r is $$\frac {4}{3} \pi r^3$$. A solid sphere of radius r that is made of a certain material weighs 40 pounds. The figure above shows half of a spherical shell made of the same material. What is the weight, in pounds, of the entire spherical shell if the outer radius is 5r and the inner radius is 2r?

A. 120
B. 360
C. 840
D. 1,080
E. 4,680

Attachment:
Untitled.png
Attachment:
GmatPrepImage.png

Hi,

We can use the Density = Mass/Volume Formula. From the Statement Density = 40/$$\frac {4}{3} \pi r^3$$
As the above figure is made up of same material Density will be same. So,

40/$$\frac {4}{3} \pi r^3$$ = x/$$\frac {4}{3} \pi (5r)^3$$ - $$\frac {4}{3} \pi (2r)^3$$

x = 4680.
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Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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VeritasPrepKarishma , Bunuel I'm not getting the concept of this problem. Could you please explain this problem with a simple approach? Thanks.
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The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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VeritasPrepKarishma , Bunuel I'm not getting the concept of this problem. Could you please explain this problem with a simple approach? Thanks. The volume of a sphere with radius r is $$\frac {4}{3} \pi r^3$$. A solid sphere of radius r that is made of a certain material weighs 40 pounds. The figure above shows half of a spherical shell made of the same material. What is the weight, in pounds, of the entire spherical shell if the outer radius is 5r and the inner radius is 2r?

A. 120
B. 360
C. 840
D. 1,080
E. 4,680

We have a sphere with radius of 5r with spherical empty space inside with radius of 2r.

The volume of a sphere with radius 5r is $$\frac {4}{3} \pi (5r)^3=125(\frac {4}{3} \pi r^3)$$.

The volume of a sphere with radius 2r is $$\frac {4}{3} \pi (2r)^3=8(\frac {4}{3} \pi r^3)$$.

The volume of solid material of the given hollow sphere = the volume of the sphere - the volume of the empty space = $$125(\frac {4}{3} \pi r^3)-8(\frac {4}{3} \pi r^3)=117(\frac {4}{3} \pi r^3)$$.

Since solid sphere of radius r having the volume of $$\frac {4}{3} \pi r^3$$, that is made of a certain material weighs 40 pounds, then our hollow sphere made of the same material and having the volume of $$117(\frac {4}{3} \pi r^3)$$, will weigh $$117*40=4,680$$.

Hope it's clear.
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Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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Bunuel For the volume of the sphere why are we taking "outer radius 5r" and "inner radius 2r" for the volume of the empty space? I can't identify it from the pic which one is the inner or outer radius, which one is solid sphere and spherical shell. Please explain these to me. Thanks.
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Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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Bunuel For the volume of the sphere why are we taking "outer radius 5r" and "inner radius 2r" for the volume of the empty space? I can't identify it from the pic which one is the inner or outer radius, which one is solid sphere and spherical shell. Please explain these to me. Thanks.

We have a sphere with radius of 5r with spherical empty space inside with radius of 2r.

Here is a cross section: Attachment: Untitled.png [ 10.27 KiB | Viewed 31978 times ]

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Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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Bunuel wrote:
VeritasPrepKarishma , Bunuel I'm not getting the concept of this problem. Could you please explain this problem with a simple approach? Thanks. The volume of a sphere with radius r is $$\frac {4}{3} \pi r^3$$. A solid sphere of radius r that is made of a certain material weighs 40 pounds. The figure above shows half of a spherical shell made of the same material. What is the weight, in pounds, of the entire spherical shell if the outer radius is 5r and the inner radius is 2r?

A. 120
B. 360
C. 840
D. 1,080
E. 4,680

We have a sphere with radius of 5r with spherical empty space inside with radius of 2r.

The volume of a sphere with radius 5r is $$\frac {4}{3} \pi (5r)^3=125(\frac {4}{3} \pi r^3)$$.

The volume of a sphere with radius 2r is $$\frac {4}{3} \pi (2r)^3=8(\frac {4}{3} \pi r^3)$$.

The volume of solid material of the given hollow sphere = the volume of the sphere - the volume of the empty space = $$125(\frac {4}{3} \pi r^3)-8(\frac {4}{3} \pi r^3)=117(\frac {4}{3} \pi r^3)$$.

Since solid sphere of radius r having the volume of $$\frac {4}{3} \pi r^3$$, that is made of a certain material weighs 40 pounds, then our hollow sphere made of the same material and having the volume of $$117(\frac {4}{3} \pi r^3)$$, will weigh $$117*40=4,680$$.

Hope it's clear.

Hello Bunuel the reason why we realised that there is an empty space inside the sphere is because of the use of word 'shell' ?
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The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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It is a straightforward question, but the visual does not make it clear that the inner sphere is empty especially when the text of the question doesn't make it apparent. I was stuck on it for a while on GMATPrepExam, then assumed that the question is asking for the difference in volume adjusted weight. It is surprising because in PS questions, the stem is generally very clear and without any ambiguity. For example, even in the same question - formula of a sphere is provided. Most of the times, PS questions don't mind giving redundant information through text that is already there in the pic.

Bunuel VeritasKarishma Have you come across other questions for geometry such as these, in which text is silent on a critical information and one has to rely on visual approach?
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Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  [#permalink]

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Sparta_750 wrote:
It is a straightforward question, but the visual does not make it clear that the inner sphere is empty especially when the text of the question doesn't make it apparent. I was stuck on it for a while on GMATPrepExam, then assumed that the question is asking for the difference in volume adjusted weight. It is surprising because in PS questions, the stem is generally very clear and without any ambiguity. For example, even in the same question - formula of a sphere is provided. Most of the times, PS questions don't mind giving redundant information through text that is already there in the pic.

Bunuel VeritasKarishma Have you come across other questions for geometry such as these, in which text is silent on a critical information and one has to rely on visual approach?

The question says that it is a spherical shell. It means it is hollow inside. A sphere is the one which is solid.
Also the question mentions outer and inner radii. So that is a hint that it is hollow inside. This is a GMAT prep question so it means you are expected to know what a spherical shell is. The diagram doesn't tell us much.
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Veritas Prep GMAT Instructor Re: The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere   [#permalink] 01 Mar 2020, 23:55

# The volume of a sphere with radius r is (4/3)*pi*r^3. A solid sphere  