Question

Using a crankshaft, a carpenter drills a hole ,1 cm in radius, through a wooden ball along a diameter. If the radius of the ball is 4 cm, what is the volume of wood remaining? .

Answer 1

Calculate the removed portion and subtract it from the original volume.
`256.3cm^3`

Explanation:

The removed portion is a cylinder with two spherical portion caps. A quick approximation could be done by ignoring the curvature. In that case the cylinder volume is just:
`V_c = pi xx r^2 xx h` ; `V_c = pi xx 1 xx 4 = 12.6 cm^3`

The Sphere volume is `V_s = 4/3pi xx r^3`
`V_s = 4/3pi xx 4^3 = 268.1 cm^3`

The remaining volume (approximated) is thus
`268.1 cm^3 - 12.6 cm^3 = 255.5cm^3`

To find the exact volume removed, add the volume of the caps:
`V_cp = pi xx h^2 xx (r – h/3)`
h = 0.127 (calculated from chord "a" - cut radius of 1 - see graphic)
`V_cp = pi xx 0.127^2 xx (4 – 0.127/3)`
`V_cp = 0.2005`
We have to take TWO of those for a total volume of 0.401
The revised cylinder length is 4 - 0.254 = 3.746
`V_c = pi xx r^2 xx h` ; `V_c = pi xx 1 xx 3.746 = 11.8 cm^3`
Add this to the caps: `11.8 + 0.401 = 12.17cm^3` (only 3.4% error from the approximation)

The remaining volume (exact) is thus
`268.1 cm^3 - 11.8 cm^3 = 256.3cm^3` (only 0.31% error from the approximation!).

http://mathworld.wolfram.com/SphericalCap.html