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Answer 1

Question:

A solid metallic sphere of diameter 28 cm is melted and recased into a number of smaller cones, each of diameter $\sf\:4\:\dfrac{3}{2}$ and height 3 cm. Find the number of cones so formed.

Answer:

The number of cones formed is 672.

Step-by-step-explanatio:

We have given the dimensions of sphere and cone.

$\sf\:For\:sphere\\\\\sf\bullet\:diameter\:(\:D\:)\:=\:28\:cm\\\\\therefore\sf\:radius\:(\:R\:)\:=\:\frac{d}{2}\\\\\implies\sf\:\cancel{\frac{28}{2}}\\\\\implies\pink{\sf\:R\:=\:14\:cm}\\\\\sf\:For\:cone\\\\\sf\bullet\:diameter\:(\:d\:)\:=\:4\:\frac{2}{3}\:cm\\\\\implies\sf\:d\:=\:\frac{14}{3}\:cm\\\\\therefore\sf\:radius\:(\:r\:)\:=\:\frac{d}{2}\\\\\implies\sf\:\dfrac{\frac{14}{3}}{ 2}\\\\\implies\sf\:\frac{\cancel{14}}{3}\:\times\:\frac{1}{\cancel2}\\\\\implies\red{\sf\:r\:=\:\frac{7}{3}\:cm}\\\\\sf\bullet\:\pink{height\:(\:h\:)\:=\:3\:cm}$

We have given that, the metallic sphere was melted and then cones were made.

We know that, volume of an object remains smae in any states.

Hence, both sphere and cone must have the same volume.

To find the number of cones formed, we have to divide the greater volume of sphere by smaller volume of cone.

Now,

$\sf\:No.\:of\:cones\:=\:\dfrac{\:Volume\:of\:sphere\:}{\:Volume\:of\:cone\:}\\\\\implies\sf\:No.\:of\:cones\:=\:\dfrac{\:\frac{4}{\cancel{3}}\:\cancel{\pi}\:r^{3}\:}{\:\frac{1}{\cancel{3}}\:\cancel{\pi}\:r^{2}\:h}\\\\\implies\sf\:No.\:of\:cones\:=\:\dfrac{\:4\:\times\:(\:14\:)^{3}}{\:(\:\frac{7}{3}\:)^{2}\:\times\:3}\\\\\implies\sf\:No.\:of\:cones\:=\:\dfrac{\:4\:\times\:\cancel{14}\:\times\:\cancel{14}\:\times\:14\:}{\frac{\cancel7}{3}\:\times\:\frac{\cancel7}{\cancel3}\:\times\:\cancel{3}}\\\\\implies\sf\:No.\:of\:cones\:=\:\dfrac{4\:\times\:2\:\times\:2\:\times\:14}{\frac{1}{3}}\\\\\implies\sf\:No.\:of\:cones\:=\:\frac{\:16\:\times\:14}{1}\:\times\:\frac{3}{1}\\\\\implies\sf\:No.\:of\:cones\:=\:224\:\times\:3\\\\\implies\boxed{\red{\sf\:No.\:of\:cones\:=\:672}}$