What percent of the box’s volume is empty? If water is poured into the box around the balls, how many ounces to the nearest 0.1 oz will it take to fill the empty space?

Question

Three balls with a diameter of 6.00 cm are dropped into a long rectangular box with a square cross section.  The balls stack up one on top of the other and they just touch the sides of the box.  The box is the right height so that the edge of the top ball is even with the top of the box.  What percent of the box’s volume is empty?  If water is poured into the box around the balls, how many ounces to the nearest 0.1 oz will it take to fill the empty space?

Answer 1

The dimensions of the box are 6×6×18 because the box has to be a ball’s diameter (6cm) in width. The area of the square base is therefore 36sq cm. The 3 balls must stack so the length must be 3 ball diameters=18cm. Volume of box=648cc.

Volume of one ball is ⁴⁄₃πr³, where r=6/2=3cm. Volume is 36πcc. Total volume of balls=108π. Volume of unfilled space=648-108π. Percentage of unfilled space=(648-108π)/648×100=(1-π/6)×100=47.64% approx.

Volume of empty space is 648-108π=308.71cc approx=308.71g because water has a density of 1 g/cc.

16oz=1lb=453.5g approx, so 308.71g=16×308.71/453.5=10.89oz, 10.9oz to 1 dec place.