Q31 of 47 Page 1

A cone of radius 10 cm is divided into two parts by a plane parallel to its base through the mid-point of its height. Compare the volumes of the two parts.

Given, radius of cone, R = 10 cm


Let height of cone be h.


This cone is divided into two parts through the midpoint of its height.


∴ AQ = AP/2


Since QD || PC,


ΔAQD ~ ΔAPC


From condition of similarity,


⇒ QD/PC = AQ/AP = AQ/2AQ


⇒ QD/R = 1/2


∴ QD = R/2


We know that volume of cone = πr2h/3


Volume of frustum = Volume of cone ABC – Volume of cone AED


⇒ (πR2H)/3 – (π/3)(R/2)2(H/2)


⇒ (πR2H)/3 – (πR2H)/24


⇒ (8πR2H)/24 – (πR2H)/24


⇒ (7πR2H)/24


Volume of part taken out/ Volume of remaining part of cone = ((πR2H)/24) / ((7πR2H)/24) = 1/7


Ans. The volume of two parts of the cone is in the ratio 1:7.


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