Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

I need help to find the interior volume?

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  • Pope
    Lv 7
    1 month ago

    Let the graph on the left be in the x-z plane, while the graph on the right is in y-z. You have described a frustum with a triangular base.

    Produce the lateral edges to meet at (-6, 0, 0). The augmented figure is then a tetrahedron. It has these vertices:

    (6, -2, 0), (6, 2, 0), (6, 0, 4), (-6, 0, 0)

    Let the base be the face lying in plane x = 6.

    base area = (4)(4)/2 = 8

    length = 12

    tetrahedron volume = (8)(12)/3 = 32

    Now, let the figure be cut by the y-z plane, and discard the part on the -x side. This leaves you with the original, given figure. The discarded part is similar to the whole, with ratio of homothety 1/2. Since it is a 3-dimensional solid, that makes the volume of the discarded part 1/8 the whole. The remaining frustum then must have 7/8 the volume of the tetrahedron computed above.

    required volume = (7/8)(32) = 28

  • 1 month ago

    The base and height are congruent and equal to (1/3)x + 2, where x runs from 0 to 6.

    The area of a typical triangle is therefore 1/2[(1/3)x + 2]^2.

    The volume will be (1/2) ∫ [(1/3)x + 2]^2 dx, where x runs from 0 to 6.

    You can easily perform this integration to get 28 cubic feet.

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