The height of a cone is 30 cm. From its topside a small cone is cut by a plane parallel to its base. If volume of smaller cone is 1/27 of the given cone, then at what height it is cut from its base?

Given height of cone, H= 30 cm

Let radius of the cone be R.



Let cone of height h be cut off from the top of the given cone and radius be r.


Consider ΔAPC and ΔAQE,


PC || QE


∴ ΔAPC ~ ΔAQE


⇒ AP/PQ = PC/QE


⇒ h/H = r/R … (1)


Given, Volume of smaller cone = 1/27 (Volume of given cone)


So, Volume of cone ABC = 1/27 (Volume of cone ADE)


Volume of cone ABC / Volume of cone ADE = 1/27


We know that volume of cone = πr2h/3


So,


⇒ (πr2h/3) / (πR2H/3) = 1/27


⇒ (r/R)2(h/H) = 1/27


From (1),


⇒ (h/H)2(h/H) = 1/27


⇒ (h/H)3 = 1/27


⇒ h/H = 1/3


⇒ h = (1/3) H


⇒ h = (1/3) (30)


∴ h = 10 cm


Now, PQ = H – h = 30 – 10


∴ PQ = 20 cm


Ans. The height at which the cone is cut from its base is 20 cm.


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