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I'm trying to solve this problem. "Use the general slicing method to find the volume of the following solids. #13. The tetrahedron (pyramid with four triangular faces), all of whose edges have length 4"

Ok So I don't see how I am suppose to use "the general slicing method" to find the volume of this solid. I had to look up the formula for volume of this solid, sense I didn't know it off the top of my head, V = (sqrt(2)a^3)/12, where a is length of the edge of one of the faces.

So I plugged in 4 for a and got (16sqrt(2))/3. I checked my answer with the back of the book to find that I got the answer correct. I got the right answer but have no idea how to solve this problem using calculus. Thank you for any help you can provide.

If I split the one of the faces, the base, in half so that way it forms a right triangle, given that one of the angles is pi/3, each face is a equilateral triangle, each face has a length of 4, the hypotenuse in this case, the shorter leg is half that 2, the longer leg must be 2sqrt(3) (from c^2=a^2+b^2), the first derivative of the equation that can be used to represent the hypotenuse is sqrt(3), and the equation that can be used to represent the hypotenuse would than be y=-sqrt(3)x+2sqrt(3). I added the negative sign because I don't feel like integrating absolute values.

Solving for x we get

x= 2 - y/sqrt(3)

If I than multiply this equation by two to get the length of the face at any individual point along the hypotenuse (the original equation was only for half the face) I get

x = 4 - (2y)/sqrt(3)

This were I get stuck. I haven't studied geometry in over 4 years, and that was just basic euclidean geometry. I'm unfamiliar with the shape of individual slices that would come out of the base along each point of base. If I knew what shape this is I could just integrate the area formula of it from 0 to 2sqrt(3) assuming the area formula for this shape is some function of height

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# Calculus I - Volume by Slicing

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