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Volume of a Sphere with a Retangluar Prism minused from Center

What is the resultant volume of a SPHERE, that has a radius of 100m, that has a rectangular prism, which is CENTERED EXACTLY WITH THE CENTER OF THE SPHERE and that measures 100m x 10m x 10m, SUBTRACTED from it?

Show how the answer was completed. This is for fun and not for homework.

Thanks in advance math experts!

phoffric

Vs = 4 pi R³ = 4 pi (100)³

Vr = 100m x 10m x 10m = 10^4

Vol = 4 pi 10^6 - 10^4 = (400 pi - 1) 10^4 =~ 4 × 10^6 × pi (m³)

cyberkiwi

Whoa... hold there a second cowboy.
At those dimensions, your prism will not stand on end because it is not flat. Remember it is cut out from the sphere...

Thanks, please try again.

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phoffric

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cyberkiwi

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cyberkiwi

phoffric

You are right you know.
I read the question and I thought it said 100m diameter, cutting a "prism" right trough it at 100m x 10m x 10m. Now that's not possible at all because the sides of the prism will never reach 100m.
But if you are just magically zapping a prism from its belly then it's no issue.

NerdsOfTech

ASKER

Thanks phoffric. I think that you are forgetting that the rectangular prism comes out of the sphere and the rectangular prism's "complete" volume should not affect the volume of the sphere.

Very nice image cyberwiki. you are on the right track.

Hint:
Sphere - (modified prism of length equal to sphere radius) - 2(sphere cap)

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phoffric

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cyberkiwi

Hi NerdsOfTech,

I don't think you worded the question correctly (wrong dimensions).

In any case, I believe the formula complete, just need to plug numbers into it - but I am on the job earning real dough so good luck.

NerdsOfTech

ASKER

Sphere - (modified prism of length equal to sphere DIAMETER) - 2(sphere cap)

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cyberkiwi

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NerdsOfTech

ASKER

OK I will add a related question next with a bigger rectangular prism outside sphere ... I know this one looks too close to the edge
Sher-RPMinus.jpg

NerdsOfTech

ASKER

oops sorry wiki. fixed: DOH!

Sphere - (modified prism of length equal to the prism with DIAMETER diagonals) - 2(sphere cap)

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phoffric

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cyberkiwi

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NerdsOfTech

ASKER

oh DOH i failed on the question. awarding points. Sorry LOL. Related question will be better.

NerdsOfTech

ASKER

phoffric you got it! Next time I'll be more careful about setting up the question.

ozo

A 100m x 10m x 10m rectangular prism fits entirely within a 100m radius sphere, an there is no problem with end caps.
If it were a 200m x 10m x 10m prism, then the volume removed would be
8*integral(x=0..5:integral(y=0..5:sqrt(100^2-x^2-y^2))
= 8*(((10*5*Sqrt[9975 - 5^2])/3 + (149875*ArcSin[5/(5*Sqrt[399])])/3 - (5*(-30000 + 5^2)*ArcTan[5/Sqrt[9975 - 5^2]])/3 + (1000000*ArcTan[(5*(-399 + 4*5)*Sqrt[9975 - 5^2])/ (-9975 + 5^2)])/3 + (1000000*ArcTan[(5*(399 + 4*5)*Sqrt[9975 - 5^2])/ (-9975 + 5^2)])/3)/2) = 19983.32359769

NerdsOfTech

ASKER

ozo I have a related question that has been fixed. Can you go there to solve? Also, isn't the related question possible to solve without integration? Thanks