Slant Cut

Slant Cut

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The cone is cut by a plane parallel to its base. The small cone on top is similar to the large cone. the ratio

of the slant heights of the cone is 1:2.
Find the ratio indicated.
a)the surface area of the large cone to that of the small cone; the volume of the large cone to that of the small cone.
b)the surface area of the frustum to that of the large cone; to that of the small cone.
c)the volume of the frustum to that of the large cone; to that of the small cone.

If the slant heights are 1:2, then :
Slant height of small cone = Slant height of frustum

Let R = radius of large cone , H = height of large cone.
Let r = radius of small cone , h = height of small cone.

By ratios in similar triangles, we have : 1/(2r) = 2/2R,
which gives : r =R/2.
Also, we have : h = H/2

General Formulae :
Lateral surface area of a cone = πRL (L = lateral length)
Volume of a cone = πR^2H/3

(a)(i) Lat. surface area of large cone = πR2
Lat. surface area of small cone = π(R/2)1
Ratio = πR2 / [π(R/2)1] = 4

(a)(ii) Volume of large cone = πR^2H/3
Volume of small cone = π(R/2)^2(H/2)/3 = πR^2H/24
Ratio = (πR^2H/3) / ( πR^2H/24) = 8

(b)(i) Lat. surface area of frustum =
Lat. S.A. of large cone - Lat. S.A. of small cone
= 2πR - πR/2 = 3πR/2
Ratio = (3πR/2) / (2πR) = 3/4

(b)(ii) Ratio = (3πR/2) / (πR/2) = 3

(c)(i) Volume of frustum =
Volume of large cone - Volume of small cone
= πR^2H/3 - πR^2H/24 = 7πR^2H/24
Ratio = (7πR^2H/24) / (πR^2H/3) = 7/8

(c)(ii) Ratio = (7πR^2H/24) / (πR^2H/24) = 7