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Question

A solid cone of base radius 10 cm is cut into two parts through the mid-point of its height, by a plane parallel to its base. Find the ratio in the volumes of two parts of the cone.

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Solution



Let the height of the cone be H.
Now, the cone is divided into two parts by the parallel plane
∴ OC = CAH2
Now, In ∆OCD and OAB
∠OCD = OAB (Corresponding angles)
∠ODC = OBA (Corresponding angles)
By AA-similarity criterion ∆OCD ∼ ∆OAB
CDAB=OCOACD10=H2×HCD=5 cm
Volume of first partVolume of second part=13πCD2OC13πCAAB2+ABCD+CD2=52102+105+52=25100+50+25=25175=17CDR=H2×HCD=R2
Volume of smaller coneVolume of whole cone=13π(CD)2OC13π(AB)2OA=(R2)H2R2H=18R2HR

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