According to the Official Rules of Baseball, a baseball must have a circumference not more than 9.25 in or less than 9.00 in and a mass not more than 5.25 oz or less than 5.00 oz. What range of densities can a baseball be expected to have? Express this range as a single number with an accompanying uncertainty limit.

To find the radius of the baseball, we can use the formula for the circumference of a circle: \(C = 2\pi r\), where \(C\) is the circumference and \(r\) is the radius. We need to find the radius for both the minimum and maximum allowed circumferences. For the minimum circumference (9.00 in): \[ r_{min} = \frac{C_{min}}{2\pi} = \frac{9.00}{2\pi} \] For the maximum circumference (9.25 in): \[ r_{max} = \frac{C_{max}}{2\pi} = \frac{9.25}{2\pi} \]

Now that we have the values for the radius of the baseball, we can calculate the volume for both minimum and maximum radius using the formula for the volume of a sphere: \(V = \frac{4}{3}\pi r^3\). For the minimum radius: \[ V_{min} = \frac{4}{3}\pi (r_{min})^3 = \frac{4}{3}\pi \left(\frac{9.00}{2\pi}\right)^3 \] For the maximum radius: \[ V_{max} = \frac{4}{3}\pi (r_{max})^3 = \frac{4}{3}\pi \left(\frac{9.25}{2\pi}\right)^3 \]

Now we can calculate the range of densities for the given mass range, using the formula for density: \(\rho = \frac{m}{V}\), where \(\rho\) is density, \(m\) is mass, and \(V\) is volume. Density for minimum mass (5.00 oz) and minimum volume (9.00 in): \[ \rho_{1} = \frac{m_{min}}{V_{min}} = \frac{5.00}{\frac{4}{3}\pi \left(\frac{9.00}{2\pi}\right)^3} \] Density for maximum mass (5.25 oz) and minimum volume (9.00 in): \[ \rho_{2} = \frac{m_{max}}{V_{min}} = \frac{5.25}{\frac{4}{3}\pi \left(\frac{9.00}{2\pi}\right)^3} \] Density for minimum mass (5.00 oz) and maximum volume (9.25 in): \[ \rho_{3} = \frac{m_{min}}{V_{max}} = \frac{5.00}{\frac{4}{3}\pi \left(\frac{9.25}{2\pi}\right)^3} \] Density for maximum mass (5.25 oz) and maximum volume (9.25 in): \[ \rho_{4} = \frac{m_{max}}{V_{max}} = \frac{5.25}{\frac{4}{3}\pi \left(\frac{9.25}{2\pi}\right)^3} \] We want to find the minimum and maximum density values from the calculated densities \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3}\), and \(\rho_{4}\).

To express the density range as a single number with an accompanying uncertainty limit, we calculate the average of the minimum and maximum densities and the difference between the maximum and minimum densities as the uncertainty limit. Average density: \[ \rho_{avg} = \frac{\rho_{min} + \rho_{max}}{2} \] Uncertainty limit: \[ \rho_{uncertainty} = \frac{\rho_{max} - \rho_{min}}{2} \] So, the density range of a baseball is \(\rho_{avg} \pm \rho_{uncertainty}\).