Question

A hollow cone is cut by a plane parallel to the base and then the upper portion is removed. so now If the csa of remainder is 8/9th of the curved surface of the whole cone, find the ratio of the line segments into which the cone's altitude is divided by the plane.​

Answer 1

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 Let the:-

⭕️ Height of the larger cone = H

⭕️Height of the smaller cone = h

⭕️ radius of the Larger cone = R

⭕️radius of the smaller circle = r

$⇒ h/H = r/R = l/L$

It's given that the :-

⭕️CSA of the frustum = (8/9) Curved surface area of the cone.

$⇒ π (R + r) (L – l) = (8/9) × π × R × L$

 $⇒ (1 + r/R) (1 – l/L) = (8/9)$

 $⇒ (1 + h/H) (1 – h/H ) = (8/9)$

⭕️Simplifying, we get h²/H² = 1/9

 $∴ h/H = 1/3$

 $∴ h/(H- h) = 1/2$

Answer 2

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Let the:-

✯Height of the larger cone = H

✯Height of the smaller cone = h

✯Radius of the Larger cone = R

✯Radius of the smaller circle = r

⇒ h/H = r/R = l/L ⇒h/H=r/R=l/L

It's given that the :-

✯CSA of the frustum = (8/9) Curved surface area of the cone.

⇒ π (R + r) (L – l) = (8/9) × π × R × L ⇒π(R+r)(L–l)=(8/9)×π×R×L

⇒ (1 + r/R) (1 – l/L) = (8/9)⇒(1+r/R)(1–l/L)=(8/9)

⇒ (1 + h/H) (1 – h/H ) = (8/9)⇒(1+h/H)(1–h/H)=(8/9)

✯Simplifying, we get h²/H² = 1/9

∴ h/H = 1/3∴h/H =1/3

∴ h/(H- h) = 1/2∴h/(H−h)=1/2