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 recasted into a third solid cylinder of height 10 cm. Find the diameter of the cylinder formed.

For new cylinder formed,

Let Radius = r

and

Volume = V

Height (h) = 10 cm (Given)

For 1st cylinder melted,

Diameter (d) = 60 cm

Radius (r₁) = 30 cm

Height (h₁) = 30 cm

Volume = V₁

For 2nd cylinder melted,

Radius (r₂) = 30 cm

Height (h₂) = 60 cm

Volume = V₂

Volume of new cylinder formed will be equal to the sum of two cylinders melted,

V = V₁ + V₂

$\Rightarrow \dfrac{1}{3}πr^2h = \dfrac{1}{3}πr$1^2h1 + \dfrac{1}{3}πr2^2h2 \\[1em] \Rightarrow r^2h = r1^2h1 + r2^2h2 \\[1em] \Rightarrow r^2 \times 10 = (30)^2 \times 30 + (30)^2 \times 60 \\[1em] \Rightarrow r^2 \times 10 = 27000 + 54000 \\[1em] \Rightarrow r^2 = \dfrac{81000}{10} \\[1em] \Rightarrow r^2 = 8100 \\[1em] \Rightarrow r = \sqrt{8100} = 90 \text{ cm.}

Diameter = 2r = 2 x 90 = 180 cm.

Hence, diameter of new cylinder = 180 cm.