Let’s take the size of the US cent (penny) to be 0.75 in in diameter, and 0.06 in thick. Since the coins are circular, each one will fit into a square 0.75²=0.5625 sq in. The volume of the square prism is 0.5625×0.06=0.03375 cu in.
The volume of the box is 16×8×9=1152 cu in, so about 1152/0.03375=34,133 pennies should fit into the box ($341.33 worth!).
However, the only dimension the coins will fit side by side exactly is 9 in=12 pennies. We can get 16/0.75=21 coins along the 16 in side but only 8/0.75=10 along the 8 in side. To be efficient, we’ll use the 8 in side for stacking, so 8/0.06=133 coins. So far we can fit 12×21×133=33516 coins. We are left with gaps into which more coins can be inserted. Along the 16 in side we have a 0.25 in gap which will fit 4 coins and, since this is one side of the rectangle 9×16, we can insert 4×12=48 coins along the 9 in side to a height of 8 in=8/0.75=10 coins, making 480 inserted coins. That brings the total to 33516+480=33,996, with no more room for any more coins. We have used 1147.365 cu in out of 1152 cu in, an efficiency of 99.6%.
