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Pyramids and cones?

Could I get any help with these two geometry-related questions? They're about solid shapes.

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  • 3 years ago
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    25.

    Pyramids (KABCD and KEFGH) are simlar.

    Ratio of volume of larger pyramid to volume of smaller pyramid

    = (V₁ + V₂) / V₁

    = (V₁ + V₁) / V₁

    = 2

    Ratio of volumes = (Ratio of sides)^3

    Assuming p is base edge of larger pyramid, then

    p/q = ∛2

    ——————————————————————————————

    26.

    H = height of cone TCD

    h = height of cone TAB

    Cones are similar

    Volume of cone TAB = v

    Volume of frustum ABCD = 26v

    Ratio of volumes

    = volume of larger cone / volume of smaller cone

    = (v + 26v) / v

    = 27

    Ratio of heights = ∛27

    H/h = 3

    H = 3h

    Height of frustum = H−h = 3h−h = 2h

    Let a = area of base AB

    Volume of cone TAB = 1/3 * a * h = ah/3

    Volume of frustum ABCD = 26ah/3

    Cylinder has base area = a and height = 2h

    Volume of cylinder = a * 2h = 2ah

    Ratio between volume of frustum and cylinder

    = (26ah/3)/(2ah)

    = 13/3

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    Note: I've tried last problem different ways, but I always end up with 13/3, which is not one of the choices listed.

    Since larger cone has volume 27 times larger than smaller cone, it must have height 3 times larger. So cone TAB has height = 1/3 of cone TCD, and frustum ABCD has height = 2/3 of cone TCD. Therefore, frustum has height 2 times that of smaller cone TAB

    If v = volume of cone TAB, then 26v = volume of frustum (given)

    Also, volume of cone = 1/3 volume of cylinder of same height and radius.

    So any cylinder with same height and base as cone TAB h volume = 3v, and cylinder with same base but twice the height has volume = 6v

    26v/(6v) = 13/3

    • I wanted to give you five stars, so sorry for the inconvenience. It was a very thorough answer, thank you a lot :D

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  • Anonymous
    3 years ago

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