Question

A solid cylinder of height 36cm and base radius 9cm is melted and recast into identical cones, each of radius 3cm and height 12 cm. Find the number of cones formed.

Answer 1

$\large\underline{\sf{Solution-}}$

• Dimensions of cylinder

$\begin{gathered}\begin{gathered}\bf\: Given-\begin{cases} &\sf{height_{(cylinder)},h = 36 \: cm} \\ &\sf{radius_{(cylinder)},r = 9 \: cm} \end{cases}\end{gathered}\end{gathered}$

So,

$\rm :\longmapsto\:Volume_{(cylinder)} = \pi \: {(radius_{(cylinder)})}^{2} \times height_{(cylinder)}$

$\rm :\longmapsto\:Volume_{(cylinder)} = \pi \: {(9)}^{2} \times 36 \: {cm}^{3} - - - (1)$

• Dimensions of Cone

$\begin{gathered}\begin{gathered}\bf\: Given-\begin{cases} &\sf{height_{(cone)},h' = 12 \: cm} \\ &\sf{radius_{(cone)},r' = 3 \: cm} \end{cases}\end{gathered}\end{gathered}$

So,

$\rm :\longmapsto\:Volume_{(cone)} = \dfrac{1}{3} \pi \: {(radius_{(cone)})}^{2} \times height_{(cone)}$

$\rm :\longmapsto\:Volume_{(cone)} = \dfrac{1}{3} \times \pi \times {(3)}^{2} \times 12$

$\rm :\longmapsto\:Volume_{(cone)} = 4\pi \times 9$

$\rm :\longmapsto\:Volume_{(cone)} = 36\pi \: {cm}^{3} - - - (2)$

Now,

Solid cylinder is melted and recast in to identical cones.

So, Number of cones are

$\rm :\longmapsto\:Number_{(cone)} = \dfrac{Volume_{(cylinder)}}{Volume_{(cone)}}$

$\rm :\longmapsto\:Number_{(cone)} = \dfrac{ \cancel\pi \: \times 9 \times 9 \times \cancel{36}}{ \cancel{36} \times \cancel \pi}$

$\rm :\longmapsto\:Number_{(cone)} =81$

Additional Information :-

• Perimeter of rectangle = 2(length× breadth)

• Diagonal of rectangle = √(length²+breadth²)

• Area of square = side²

• Perimeter of square = 4× side

• Volume of cylinder = πr²h

• T.S.A of cylinder = 2πrh + 2πr²

• Volume of cone = ⅓ πr²h

• C.S.A of cone = πrl

• T.S.A of cone = πrl + πr²

• Volume of cuboid = l × b × h

• C.S.A of cuboid = 2(l + b)h

• T.S.A of cuboid = 2(lb + bh + lh)

• C.S.A of cube = 4a²

• T.S.A of cube = 6a²

• Volume of cube = a³

• Volume of sphere = 4/3πr³

• Surface area of sphere = 4πr²

• Volume of hemisphere = ⅔ πr³

• C.S.A of hemisphere = 2πr²

• T.S.A of hemisphere = 3πr²