Question

From a wooden cone of height 20 cm, Slant height 25 cm, a hemisphere of maximum size is carved out. Radius of sphere is

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Radius\:of\:sphere=15\:cm}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies Height \: of \: cone = 20 \: cm \\ \\ \tt: \implies Slant \: height\: of \: cone = 25 \: cm \\ \\ \red{\underline \bold{To \: Find:}}\\ \tt: \implies Radius \: of \:sphere =?$

• According to given question :

$\tt \circ \: Hypotenuse = Slant \: height \\ \\ \tt \circ \: Perpendicular = Height \\ \\ \tt \circ \:Base = Radius \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies {h}^{2} = {p}^{2} + {b}^{2} \\ \\ \tt: \implies {(slant \: height)}^{2} = {(height)}^{2} + {(radius)}^{2} \\ \\ \tt: \implies {25}^{2} = {20}^{2} + {r}^{2} \\ \\ \tt: \implies 625 = 400 + {r}^{2} \\ \\ \tt: \implies 625 - 400 = {r}^{2} \\ \\ \tt: \implies 225 = {r}^{2} \\ \\ \tt: \implies r = \sqrt{225} \\ \\ \green{\tt: \implies r = 15 \: cm} \\ \\ \green{ \tt \therefore Radius \: of \:sphere \:is \: 15 \: cm}$

Answer 2

QUESTION :

From a wooden cone of height 20 cm, Slant height 25 cm, a hemisphere of maximum size is carved out. Radius of sphere is ______...

SOLUTION :

By Pythagoras Theorem, we can state that :

Hypotenuse ^2 = Perpendicular^2 + Slant Height ^2

Here the perpendicular refers to the Height of the cone and the Base refers to the base radii of the required cone.

So we can state the following :

r^2 + 20^2 = 25^2

=> r^2 = 25^2 - 20^2

=> r^2 = 625 - 400

=> r^2 = 225

=> r = 15 cm as the radius can't be negative...

∴ The Radius of the sphere is 15 cm.