Riddle: A boy leaves a cake pan outside in a sandbox as a thunderstorm approaches. It rains and the spherical raindrops completely fill up the pan. A third of the raindrops are 2 millimeter in diameter, a third of the raindrops are 3 millimeters in diameter and a third of the raindrops are 4 millimeters in diameter. The cake pan is 2 inches high and has a diameter of 12 inches. How many raindrops does it take to fill up the cake pan?
Answer to Riddle: First we need to determine the volume of each rain size and the volume of the cake pan.
Volume of cake pan = PI*radius^2*height = PI*6 inches^2*2 inches = 226.195 inches^3
226.195 inches^3 *(25.4^3 mm^3 / 1 inch^3) = 3,706,671.941 mm^3
Volume of a sphere is 4/3*PI*r^3
Volume of 2 mm rain drops = 4/3*PI*1.0mm^3 = 4.189 mm^3
Volume of 3 mm rain drops = 4/3*PI*1.5mm^3 = 14.137 mm^3
Volume of 4 mm rain drops = 4/3*PI*2.0mm^3 = 33.510 mm^3
x = number of drops for each size
x is the same number of drops for each size (1/3rd each)
x*(4.189 mm^3) + x*(14.137 mm^3) + x*(33.510 mm^3) = 3,706,671.941 mm^3
51.836*x = 3,706,671.941
x = 3,706,671.941 / 51.836 = 71,508 drops of each size
Total number of drops = 71,508 * 3 = 214,523 drops