Question

A solid right circular cylinder is made by melting a solid right circular cone. The radii of both are equal. If the height of the cone is 15cm., then let us determine the height of the solid cylinder.
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Answer 1

Answer:

solid cylinder of diameter 12cm and height 15cm is melted and recast into toys with the shape of a right circular cone mounted on a hemisphere of radius 3cm. If the height of the toy is 12cm, find the number of toys so formed.

Answer 2

$\;\star\;{\underline{\boxed{\bf{\pink{\;Answer\;}}}}}\;\star\;$

$\boxed{\sf Height \ of \ Cylinder \ = \ 5 \ cm}$

★ Diagrams :-

➸ Solid Right Circular Cylinder :

$\setlength{\unitlength}{1 cm}\begin{picture}(0,0)\qbezier(-1,0)(0,1)(1,0)\qbezier(-1,0)(0,-1)(1,0)\put(-1,0){\line(0,1){2}}\put(1,0){\line(0,1){2}}\qbezier(-1,2)(0,1)(1,2)\qbezier(-1,2)(0,3)(1,2)\put(1.5,0){\vector(0,1){2}}\put(1.5,0){\vector(0,-1){0.3}}\put(1.7,0.6){ \bf h }\put(0,0){\vector(1,0){1}}\put(1,0){\vector(-1,0){1}}\put(0.3,0.1){ \bf r }\end{picture}$

➸ Solid right circular cone :

$\setlength{\unitlength}{30} \begin{picture}(10,6) \linethickness{1.2} \qbezier(1,1)(3., 0)(5,1)\qbezier(1,1)(3.,2)(5,1)\put(3,1){\circle*{0.15}}\put(3,1){\line(0,1){3}}\qbezier(1,1)(1,1)(3,4)\qbezier(5,1)(3,4)(3,4)\put(3,1){\line(1,0){2}}\put(3.2,1.1){ \bf r \: cm }\put(1.9,1.9){ \bf 15 \: cm }\put(4,3.5){\boxed{ \bf @MagicalMystery }}\end{picture}$

★Given :

• A solid right circular cylinder is made by melting a solid right circular cone.
• The radius of both are equal.
• The height of the cone is 15cm.

★ To Find :

• The height of the solid cylinder.

★ Solution :

Let the height of cylinder be h cm.

And the radius of both cylinder and cone is r cm.

The height of the cone is 15 cm.

➸ Formulas :

$\boxed{\sf Volume \ Of \ Cylinder = \pi r^2h}$

$\boxed{\sf Volume \ Of \ Cone \ = (\pi r^2h)/3}$

According to the question,

$\boxed{\sf Volume \ Of \ Cylinder = Volume \ Of \ Cone}$

$\therefore{\pi r^2h = (\pi r^2 \times 15)/3}$

or, h = 15 ÷ 3 =5

$\boxed{\sf Height \ of \ Cylinder \ = \ 5 \ cm}$