Given :
The ball of a ballpoint pen is 1 mm in diameter and has an ASTM grain size of 10.
To Find :
How many grains are there in the ball.
Solution :
Volume of ball is :
[tex]V=\dfrac{4\pi r^3}{3}\\\\V=\dfrac{4\pi (0.5)^3}{3}\ mm^3\\\\V=\dfrac{4\times 3.14\times 0.5^3}{3}\ mm^3\\\\V=0.523\ mm^3[/tex]
Grains per unit volume is given by :
[tex]N = 2^{n-1}\times 10^3\ grains/mm^3[/tex] ( Here, n is ASTM grain size ).
Putting n = 10 , we get :
[tex]N = 2^{10-1}\times 10^3\ grains/mm^3\\\\N = 5.2\times 10^5\ grains/mm^3[/tex]
Number of grains are :
[tex]n = V\times N\\\\n = 0.523\times 5.2\times 10^5\ grains\\\\n = 2.7196\times 10^5\ grains[/tex]
Hence, this is the required solution.
