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प्रश्न
If a cone of radius 10 cm is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base. Compare the volumes of the two parts.
उत्तर
Consider a cone of radius R and height H.
Let a cone of height `H/2` is cutout from this cone whose base is parallel to the original cone. Now is
`Delta AQD and Delta APC, QD || PC`
`Delta AQD ∼ Delta APC`
`(QD)/(PC) = (AQ)/(AP)`
`QD/2 = H/(2/H)`
`QD = R/2`
Volume of cone ABC `=1/3 pi R^2 H`
Volume of cone ADE
`=1/3 pi (R/2)^2 (H/2)`
`=1/8[1/3 piR^2H]`
`=1/8`volume of cone ABC
Now volume of remaining frustum of cons EDCB
= volume of cone ABC − volume of cone AED
\[= \frac{7}{8}(\text { volume of cone ABC})\]
Comparing the volume of cone AED and frustum EDBC we get, the ratio 1 : 7.
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