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An hollow cone is cut by a plane paralel to the base and the upper portion is removed. curved surface area of the remainder is 8/9 of the curved surface of

the whole cone.find the ratio of the line segment into which the cones altitude is divided by the plane



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Let for the whole cone,
radius = R
height = H
slant height = L

for the removed of the cone,(which is also a cone)
radius = r
height = h
slant height = l

Since they are from same cone,  \frac{r}{R} = \frac{l}{L} = \frac{h}{H}

CSA of remainder = 8/9 of original
so CSA of removed portion = 1- 8/9 = 1/9 of original

CSA of original cone = πRL
CSA of removed cone = πrl

 \frac{ \pi rl}{ \pi RL} = \frac{1}{9}  \\  \\  \frac{r^2}{R^2}= \frac{1}{9}   \\  \\  \frac{r}{R} = \frac{1}{3}  \\  \\  \frac{h}{H} = \frac{1}{3}

So the cones altitude is divided in the ratio  \frac{1}{3} .

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