Question

Two solid cylinders, one with diameter 60cm and height 30cm and the other with radius 30cm and height 60cm, are melted and recast into a third solid cylinder of height 10cm. Find the diameter of the cylinder formed.

Answer 1

Hint: First, find the radius of the first cylinder by the formula, $r = \dfrac{D}{2}$. After that find the volume of the first cylinder. Then find the volume of the second cylinder. Then add the volume of both cylinders. Now equate the value with the volume of the third cylinder and substitute the values and do simplification to get the radius of the cylinder formed.

Complete step-by-step solution:
Two solid cylinders, one with a diameter of 60cm and a height of 30cm and the other with a radius of 30cm and a height of 60cm.
The radius of the first cylinder is given by,
$ \Rightarrow r = \dfrac{D}{2}$
Substitute the value of diameter,
$ \Rightarrow r = \dfrac{{60}}{2}$
Divide numerator by denominator,
$ \Rightarrow r = 30$
So, the radius of the first cylinder is 30 cm.
Now, the volume of the first cylinder is,
$ \Rightarrow {V_1} = \pi {r^2}h$
Substitute the values,
$ \Rightarrow {V_1} = \pi {\left( {30} \right)^2}\left( {30} \right)$
Simplify the terms,
$ \Rightarrow {V_1} = 27000\pi $
Now, the volume of the second cylinder is,
$ \Rightarrow {V_2} = \pi {r^2}h$
Substitute the values,
$ \Rightarrow {V_2} = \pi {\left( {30} \right)^2}\left( {60} \right)$
Simplify the terms,
$ \Rightarrow {V_2} = 54000\pi $
As both cylinders are melted, so add the volume of both cylinders,
$ \Rightarrow {V_1} + {V_2} = 27000\pi + 54000\pi $
Add the terms on the right side,
$ \Rightarrow {V_1} + {V_2} = 81000\pi $ ….. (1)
Then it is recast into a third solid cylinder. So, the volume of the third cylinder is,
$ \Rightarrow {V_3} = \pi {r^2}h$ ….. (2)
Now, equate equation (1) and (2),
$ \Rightarrow \pi {r^2}h = 81000\pi $
As the height of the third cylinder is 10 cm. Substitute the value in the above equation,
$ \Rightarrow \pi {r^2}\left( {10} \right) = 81000\pi $
Divide both sides by $10\pi $,
$ \Rightarrow {r^2} = 8100$
Take the square root on both sides,
$\therefore r = 90cm$
As we know,
$D = 2r$
Substitute the value,
$ \Rightarrow D = 2 \times 90$
Multiply the terms,
$ \Rightarrow D = 180cm$

Hence, the diameter of the cylinder formed is 180 cm.

Note: In these types of questions, the main idea is to balance the entity before starting any process and after ending the process. It is like balancing the equation. In the above question, we calculated the volume before melting. So, the volume should remain the same after melting is the key idea for solving the question.