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.
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
$\therefore \frac{\mathrm{CD}}{\mathrm{AB}}=\frac{\mathrm{OC}}{\mathrm{OA}}$
$\Rightarrow \frac{\mathrm{CD}}{10}=\frac{H}{2 \times H}$
$\Rightarrow \mathrm{CD}=5 \mathrm{~cm}$
$\frac{\text { Volume of first part }}{\text { Volume of second part }}=\frac{\frac{1}{3} \pi(\mathrm{CD})^{2}(\mathrm{OC})}{\frac{1}{3} \pi \mathrm{CA}\left[(\mathrm{AB})^{2}+(\mathrm{AB})(\mathrm{CD})+\mathrm{CD}^{2}\right]}$
$=\frac{(5)^{2}}{\left[(10)^{2}+(10)(5)+5^{2}\right]}$
$=\frac{25}{100+50+25}$
$=\frac{25}{175}$
$=\frac{1}{7}$
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