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You are trying to figure out how many gumballs you need to fill a 7.3 x 5.0 x 9.4 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.

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francocanacari

According to the calculations made, 410 gumballs will be needed to fill the box.

Since you are trying to figure out how many gumballs you need to fill a 7.3 x 5.0 x 9.4 rectangular box for Halloween, and 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, to determine this amount the following calculation must be performed:

  • (Volume of box x 5/8) / volume of gumballs = Amount of gumballs  
  • Volume of a sphere = 4/3 x 3.14 x (radius x radius x radius)  
  • Volume of a gumball = 4/3 x 3.14 x (0.5 x 0.5 x 0.5) = 0.5235 inches  
  • ((7.3 x 5 x 9.4) x 5/8) / 0.523 = X
  • (343.1 x 5/8) / 0.523 = X
  • 214.4375 / 0.523 = X
  • 410 = X

Therefore, 410 gumballs will be needed to fill the box.

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