Topic: CalculusMany types of sports balls — tennis, racquetball, wiffle balls to name a few — are packed in cylindrical cans, and since the balls are round, a lot of empty space is left in the can. How much space?So as to determine the extent of the void in a cylindrical can containing spherical sports balls it is necessary to first determine the volume of the cylindrical can and then to this subtract the volume of the balls. Suppose the height of the cylindrical can be \( h \) while the radius that of the can be \( R \). The volume \( V_{can} \) of the cylindrical can is given by:From the dimensions provided, the volume of the cylindrical can is expressed by the following: \[ Thus it is convenient to define can volume as V_{can} = \pi R^2 h \] If the can holds \( n \) balls, each with radius \( r \), the total volume occupied by the balls \( V_{balls} \) can be calculated using the formula for the volume of a sphere:If the can can take \( n \) balls of radius \( r \) each, then the total volume occupied by the balls \( V_{balls} \) can easily be determined using the formula and equation of a sphere that is \(V_{balls} = \frac{4}{3}\pi r^3 n \). To find the volume of a solid ball the following is employed \[ V_{ball} = \frac{4}{3} \pi r^3 \] \[ V_{balls} = balls \times \frac{4}{3} \pi \times r^3 \] To find the empty space, subtract the total volume of the balls from the volume of the can:Thus, the empty space can be calculated as follows: Volume of the can minus the total volume of the balls. \[ V = V_{can} – V_{balls} \] That is \[ V_{empty} = \pi R^2 h – n \times \frac{4}{3} \pi r^3 \] The balls as real objects are not perfect hence, they cannot fit the interior of the cylindrical can like perfect spheres, thus the constant gap. It is through the arrangement that one is able to get efficiency in packing or in this case the way the spheres fit the can. Typically, in many practical situations, packing efficiency of spheres in a cylinder lies between 64 and 74 per cent, implying that there are approximately 26 to 36 per cent of voids.
