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Math Help - The base of a solid is a circle of radius a , and every plane

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    The base of a solid is a circle of radius a , and every plane

    The base of a solid is a circle of radius a, and every plane section perpendicular to a diameter is a square. The solid has volume:
    (A) \frac{8}{3}a^3

    (B) 2\pi a^3

    (C) 4\pi a^3

    (D) \frac{16}{3}a^3

    (E) \frac{8\pi}{3}a^3

    .
    Last edited by yoman360; April 20th 2010 at 05:20 PM.
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    Quote Originally Posted by yoman360 View Post
    The base of a solid is a circle of radius a, and every plane section perpendicular to a diameter is a square. The solid has volume:
    (A) \frac{8}{3}a^3

    (B) 2\pi a^3

    (C) 4\pi a^3

    (D) \frac{16}{3}a^3

    (E) \frac{8\pi}{3}a^3

    .
    Place the circle in a coordinate system with the center at the origin. The equation of the circle is

    x^2+y^2=a^2

    y=\sqrt{a^2-x^2}

    Each slice perpendicular to the x-axis can be approximated by a right rectangular prism with dimensions 2y by 2y by dx.

    The volume V of the prism is

    V=(2\sqrt{a^2-x^2})(2\sqrt{a^2-x^2})\,dx

    Integrate from -a to a to get the volume of the solid
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  3. #3
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    Not too sure if I am correct here, but is that not just describing a cylinder of height equal to diameter? After all, if you take the cross-section across the diameter of a cylinder then don't you just get a rectangle?
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    Quote Originally Posted by Diemo View Post
    Not too sure if I am correct here, but is that not just describing a cylinder of height equal to diameter? After all, if you take the cross-section across the diameter of a cylinder then don't you just get a rectangle?
    If the solid were a cylinder, every plane section perpendicular to a diameter is a rectangle but not necessarily a square.
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    ok i'll try it
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    All a square is is a rectangle with sides the same length. The more that I think about it the more sure I am that the answer is is merely the volume of a cylinder with height 2a. Which is...
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  7. #7
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    Quote Originally Posted by ione View Post
    Place the circle in a coordinate system with the center at the origin. The equation of the circle is

    x^2+y^2=a^2

    y=\sqrt{a^2-x^2}

    Each slice perpendicular to the x-axis can be approximated by a right rectangular prism with dimensions 2y by 2y by dx.

    The volume V of the prism is

    V=(2\sqrt{a^2-x^2})(2\sqrt{a^2-x^2})\,dx

    Integrate from -a to a to get the volume of the solid
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    Ione

    I have been thinking about this for the past while. Basically, what you did is you put the the base in the z=0 plane, and then uintegrated over x, using y^2=a^2-x^2 with the volume of each slice being 2y by 2y by dx. Which as I see it is correct, provided that each slice is the same height. Each slice has to have the same height by symmetrtry, as It doesn't matter when you rotate the object. This gives the total volume as \frac{16}{3}a^3.

    However, the volume of a cylinder is \pi r^2 h. Taking our h to be twice the radius (2r) then gives the volume as 2*pi*a^3, which means that it has two different volumes, or something that I am thinking of is wrong. Is the object described not a cylinder? Which actually doesn't change the problem, because when you use your method to find the cylinder volume, you get 23*y*hdx, which still does not give any pi dependance. So what am I missing?
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    Quote Originally Posted by Diemo View Post
    Ione

    I have been thinking about this for the past while. Basically, what you did is you put the the base in the z=0 plane, and then uintegrated over x, using y^2=a^2-x^2 with the volume of each slice being 2y by 2y by dx. Which as I see it is correct, provided that each slice is the same height. Each slice has to have the same height by symmetrtry, as It doesn't matter when you rotate the object. This gives the total volume as \frac{16}{3}a^3.

    However, the volume of a cylinder is \pi r^2 h. Taking our h to be twice the radius (2r) then gives the volume as 2*pi*a^3, which means that it has two different volumes, or something that I am thinking of is wrong. Is the object described not a cylinder? Which actually doesn't change the problem, because when you use your method to find the cylinder volume, you get 23*y*hdx, which still does not give any pi dependance. So what am I missing?
    The problem states "every plane section perpendicular to a diameter is a square." Since the length of the slices are changing, the height of the solid is changing. A cylinder has constant height, so the solid is not a cylinder.
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    It claims every plane perpendicular to a diameter is a square. This means that the height has to be the same at every point surely? By symmettry?

    Consider first the diameter in the x direction. The plane perpendicular to that diameter is a square. Now consider the diameter in the y direction, the plane perpendicular to that diameter is a square. Now consider the diameter in the 45 degree direction, the plane perpendicular to that diameter is a square, furthermore, it passes through the two previous planes, hence the squares are all the same height. Wrong in some way?
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    Quote Originally Posted by Diemo View Post
    It claims every plane perpendicular to a diameter is a square. This means that the height has to be the same at every point surely? By symmettry?

    Consider first the diameter in the x direction. The plane perpendicular to that diameter is a square. Now consider the diameter in the y direction, the plane perpendicular to that diameter is a square. Now consider the diameter in the 45 degree direction, the plane perpendicular to that diameter is a square, furthermore, it passes through the two previous planes, hence the squares are all the same height. Wrong in some way?
    "The base of a solid is a circle of radius a, and every plane section perpendicular to a diameter is a square."

    I took this to mean every plane section perpendicular to one specific diameter is a square.
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  12. #12
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    Quote Originally Posted by ione View Post
    "The base of a solid is a circle of radius a, and every plane section perpendicular to a diameter is a square."

    I took this to mean every plane section perpendicular to one specific diameter is a square.
    Ahh, yes, think your right actually. Sorry for the bother.
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