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  1. #1
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    tetrahedron

    Calculate the volume of the tetrahedron limited by the plan n1: 3x+2y-4z-12=0 and plans for coordinated.



    Answer:
    12
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    Quote Originally Posted by Apprentice123 View Post
    Calculate the volume of the tetrahedron limited by the plan n1: 3x+2y-4z-12=0 and plans for coordinated.



    Answer:
    12
    I think you are asking for the volume of the tetrahedron bounded by n_1 and the coordinate planes?

    If so, one vertex will be at the origin, and the other three will be located at the points where the plane cuts the x, y,\text{ and }z axes, namely at (4,0,0),\;(0,6,0),\text{ and }(0,0,-3).

    Thus, the volume is

    V = \pm\frac16\left\lvert\begin{matrix}<br />
0 & 0 & 0 & 1\\<br />
4 & 0 & 0 & 1\\<br />
0 & 6 & 0 & 1\\<br />
0 & 0 & -3 & 1<br />
\end{matrix}\right\rvert = \frac16\cdot72 = 12
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  3. #3
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    Hello, Apprentice123!

    The volume of a pyramid with base area B and height h is: . V \;=\;\frac{1}{3}Bh


    Calculate the volume of the tetrahedron limited by the plane n_1:\;\;3x+2y-4z-12\:=\:0
    and the coordinate planes.
    The vertices are: . (0,0,0),\:(4,0,0),\;(0,6,0,),\:(0,0,-3)

    The base is a right triangle with legs 4 and 6.
    . . Its area is: . B \:=\:\frac{1}{2}(4)(6) \:=\:12\text{ units}^2

    The height is: . h \,= \,3\text{ units}

    Therefore: . V \;=\;\frac{1}{3}(12)(3) \;=\;12\text{ units}^3

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  4. #4
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    Thank you for the help of all
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