Question

A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use \(\pi = \frac{22}{7}\))

Hint:

Always group your \(\pi r^2\) terms before calculating. It saves you from multiplying by \(22/7\) multiple times and reduces rounding errors.

Answer 1

Step 1: Understanding the Structure of the Solid:
The solid is made up of:
• One cylindrical portion in the middle
• Two hemispherical ends

Since two hemispheres together form one complete sphere,
Total Volume = Volume of Cylinder + Volume of Sphere

Given:
Total height of solid = 20 cm
Diameter = 7 cm
So radius r = 7/2 = 3.5 cm

Step 2: Finding Height of the Cylindrical Part:
The hemispheres contribute height equal to 2r.

Height of cylinder h = Total height − 2r
= 20 − 7
= 13 cm

Step 3: Applying Volume Formulae:
Volume of cylinder = πr²h
Volume of sphere = (4/3)πr³

So,
V = πr²h + (4/3)πr³

Take πr² common:
V = πr² (h + (4/3)r)

Substitute r = 3.5 and h = 13:
V = (22/7) × (7/2) × (7/2) × (13 + (4/3 × 7/2))

Simplify step by step:
(22/7) × (49/4) × (13 + 14/3)

= (22 × 49) / 28 × ( (39 + 14)/3 )
= 38.5 × (53/3)

= (38.5 × 53) / 3
= 680.17 cm³ (approx.)

Final Answer:
The total volume of the solid is approximately 680.17 cm³.