Question

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

Answer 1

Here, first find the volumes of the two cylinders using the formula of volume of cylinder of given radius and height. Add the two volumes obtained, and then equate this volume to the volume of the third cylinder. Now, we have one equation and one variable i.e., radius of the third cylinder, which can be obtained by solving the equation. Multiply the radius obtained by 2 to get its diameter.

Complete step by step explanation:
We are given two cylinders,
Cylinder 1: Diameter = 60 cm ⇒ Radius, R = 30 cm and Height, H = 30 cm
Volume of cylinder = $\pi {R^2}H = \pi \times 30 \times 30 \times 30 = 27000\pi$ cu. cm
Cylinder 2: Radius, r = 30 cm and Height, h = 60 cm
Volume of cylinder = $\pi {r^2}h = \pi \times 30 \times 30 \times 60 = 54000\pi$ cu. cm
Now, Sum of the volumes of the two cylinders $= 27000\pi + 54000\pi = 81000\pi$ cu. cm
Let the radius of the third cylinder be x and given that the height of the third cylinder is 10 cm.
We know that, in conversion of shapes of solid, its volume remains the same.
Applying this concept and comparing the sum of volume of two cylinders with the third cylinder, we get
$\pi {x^2} \times 10 = 81000\pi \Rightarrow {x^2} = 8100 \Rightarrow x = 90$ cm

Hence, the radius of the third cylinder is 90 cm.
Diameter = 2 × radius = 2 × 90 cm = 180 cm

Note: In these types of questions, we use the concept of conversion of solid shapes in which volume remains unchanged. Keep in mind their surface area will change but volume remains the same. We can change the cylinder to cube, cuboid etc, but the basic quantity i.e. volume is fixed.