Question

the height of the cone is 10cm. the cone is divided into two parts using a plane parallel to its base at the middle of its height. find the ratio of the volumes of the parts.​

Answer 1

$\bold\red{\boxed{Question}}$

The height of the cone is 10cm. the cone is divided into two parts using a plane parallel to its base at the middle of its height. find the ratio of the volumes of the parts.

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$\mathfrak\red{\boxed{Solution}}$

$Let \:the \:height \:of \:the \:given\: cone = h$ cm

$after\: dividing \:in \:two \:parts,\: we \:get$

(i) $frustum\: of\: the\: cone \:with\: radius, R=10cm$

and $radius\: r = 5cm, height = \dfrac{h}{2}$cm

(ii) $a \:smaller \:cone\: of \:radius, r =5cm\: and\: height=10$

∴ $Ratio\: of \:volumes = \dfrac{volume\: of\: frustum \:of \:cone}{volume \:of smaller\: cone}$

= $\dfrac{\dfrac{1}{3}πr^{2}(\dfrac{h}{2})}{\dfrac{1}{3}π(\dfrac{h}{2})[R^{2}+ r^{2} + Rr]}$

= $\dfrac{5 × 5}{10^{2} + 5^{2} + 10 + 5}$

= $\dfrac{25}{175}$

= $\dfrac{1}{7}$

=$1 : 7$

$1:7 \:is \:the \:required\: Ratio$