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Math Help - Find the volume of the described solid.

  1. #1
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    Find the volume of the described solid.

    A pyramid with height h and base an equilateral triangle with side a (a tetrahedron).

    This is an example problem from the text. If someone could walk me through how to do this problem using integration, I think it will help me grasp how to do the other problems.

    I'm having trouble starting, and not sure what I'm supposed to do. I think I'm supposed to find the surface area of the triangle with respect to height, and evaluate it from 0 to h. But how to I find out what the triangular base is at any given height?
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  2. #2
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    Cross-Sectional Area

    Imagine the center of the equilateral triangle base sitting on the origin in the xy-plane. The figure comes to a point somewhere along the z-axis at the point (0,0,h).

    The general formula for finding volume of such a figure is V=\int_a^bA(r)dr, where A(r) is the cross-sectional area of the figure at a distance r from the origin. Here, your bounds of integration will be 0\to h. A(h)=0 since it comes to a point, and A(0)=\frac{\sqrt3}2a^2 since this is the area of an equilateral triangle with side length a.

    Since the side-length of the equilateral cross-sectional triangle shrinks linearly from a to 0, it can be represented as s(r)=\frac{h-r}{h}a, so the area A(r)=\frac{\sqrt3}2a^2\frac{(h-r)^2}{h^2}

    And therein lies your integral: V=\frac{\sqrt3}2\frac{a^2}{h^2}\int_0^h(h-r)^2dr

    (You can of course check your work by using the simple geometric formula V=\frac13bh for figures that start at a base of area b and rise to a point at height h.)
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  3. #3
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    Illinois
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    Thumbs up Wonderful answer

    This is exactly what I needed to hear. It all makes sense now. I only wish I had gotten the reply before I had to go into class an hour and a half ago. Oh well, at least I understand it better now.
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