Question

The given figure represents a solid consisting of a cylinder surmounted by a cone at one end and a hemisphere at the other. Find the volume of the solid.

Answer 1

Given:

Cylinder diameter = 7 cm ⇒ radius r = 3.5 cm.

Cylinder length = 6.5 cm.

Total length from the cylinder’s left face to the cone tip = 12.8 cm, so cone height = 12.8 – 6.5 = 6.3 cm.

A hemisphere of radius 3.5 cm is attached to the other end.

Step-wise calculation:

1. Volume of the cylinder V_cyl = πr²h 

= π(3.5)²(6.5)

= π × 12.25 × 6.5

= `(637/8)π cm^3`

2. Volume of the cone `V_"cone" = (1/3)πr^2h` 

= `(1/3)π(3.5)^2(6.3)`

= `(3087/120)π cm^3`

3. Volume of the hemisphere `V_"hemi" = (2/3)πr^3`

= `(2/3)π(3.5)^3`

= `(343/12) πcm^3`

4. Total volume V_total = V_cyl + V_cone + V_hemi

= `(637/8 + 3087/120 + 343/12) π`

= `(2009/15) πcm^3`

Numeric approximation (using π ≈ 3.14159265):

V_total ≈ 133.933333... × π ≈ 420.8 cm³.

The volume of the solid is `(2009/15)` π cm³ ≈ 420.8 cm³.