AMDM Question (Mathematics)

Question

You are trying to figure out how many gumballs you need to fill a 4.1 x 5.8 x 6.9 rectangular box for Halloween. Each gumball has a radius of 1/2 in, if the packing density for spheres is 5/8 of the volume will be filled with gumballs while the rest will be air how many gumballs will be needed? Round to the nearest whole number.

Answer 1

The volume is 4.1 * 5.8 * 6.9 = 164.082

5/8 will be filled with gumballs or 102.551 volume

each gumball (sphere) has a volume of 4/3 * PI * radius ^3 = 4/3 * 3.14159 * (1/2)^3 = 0.524

number of gumballs = volume for gumballs/ volume each gumball = 102.551/0.524 = 195.7 ~ 196 gumballs