Question

A cylinder and a cone have equal bases. The height of the cylinder is 3 cm and the area of its base is 100 cm². The cone is placed upon the cylinder. The volume of the solid figure so formed is 500 cm³. Find the total height of the figure.

Answer 1

Given,
Height of the cylinder (h) = 3 cm
Area of the base of cylinder = 100 cm²
Volume of the solid figure = 500 cm³

A cylinder and a cone have equal bases.
∴ They have equal radii.
Radius of cylinder = Radius of cone = Area of base = 100 cm² 
∴ πr² = 100         ...(i)

Let the height of the cone be H cm.

Volume of the solid figure = Volume of the cylinder + Volume of the cone

∴ 500 = πr²h + ${\displaystyle \frac{1}{3}}$πr²H

∴ 500 = πr² ${\displaystyle (\text{h}+\frac{\text{H}}{3})}$

∴ 500 = 100 ${\displaystyle (3+\frac{\text{H}}{3})}$        ...[From (i)]

∴ ${\displaystyle 3+\frac{\text{H}}{3}=\frac{500}{100}}$

∴ ${\displaystyle 3+\frac{\text{H}}{3}=5}$

∴ ${\displaystyle \frac{\text{H}}{3}=5-3}$

∴ ${\displaystyle \frac{\text{H}}{3}=2}$

∴ H = 2 × 3

∴ H = 6 cm

∴ Height of conical part (H) = 6 cm

Total height of the figure = h + H = 3 + 6 = 9 cm

Thus, the total height of the figure is 9 cm.