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Calculus

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A ball of radius 12 has a round hole of radius 6 drilled through its center. Find the volume of the resulting solid.

I tried finding the volume of the sphere and the volume of the cyclinder then subtract however that did not work.

  • Calculus -

    the part that is cut out,what you call a "cylinder" is not really at cylinder.
    you are forgetting about the caps on each end of your 'cylinder'

    we will have to use Calculus to do that
    Visualize a circle, centre at the origin and radius of 12,rotating about the x-axis resulting in our sphere.

    NOw visualize a drill bit of radius 3 as the x-axis, drilling out a hole.

    volume of sphere = (4/3)pi(12)^3 = 7238.229
    (you probably got that)


    now the 'cylinder will cut at (√135,3)and (-√135,3)
    so the volume of the cylinder with flat tops = pi(3)^2(2(√135)) = 1642.59

    ( I am going to assume you got an answer of 7238.229-1642.59 = 5595.639)

    I will calculate one of the "caps", then subtract twice that from the above answer.
    vol. of cap = pi[integral](144-x^2)dx from √135 to 12
    = pi[144x - (1/3)x^3│ from √135 to 12
    = 5.4159
    CHECK MY ARITHMETIC, THIS IS WHERE I USUALLY SCREW UP

    so total volume
    = 7238.229 - 1642.59 - 2(5.4159
    = 5584.8072

  • silly arithmetic error Calculus -

    ARRGGGHH! ARITHMETIC ERROR!!

    << so the volume of the cylinder with flat tops = pi(3)^2(2(√135)) = 1642.59 >>

    should have said:
    so the volume of the cylinder with flat tops = pi(3)^2(2(√135)) = 657.036

    and then

    <<( I am going to assume you got an answer of 7238.229-1642.59 = 5595.639) >>

    should say:

    ( I am going to assume you got an answer of 7238.229-657.036 = 6581.19

    and finally at the end
    << so total volume
    = 7238.229 - 1642.59 - 2(5.4159
    = 5584.8072 >>

    should say:

    so total volume
    = 7238.229 - 657.036 - 2(5.4159
    = 6570.36

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