To calculate the mass of a typical black hole.
This information: If gas and stars can be observed orbiting a black hole at a velocity of v kilometres per second and if the orbiting objects are at a distance of r light years from the black hole, then the mass of the black hole in units of solar masses (M) can be calculated with the formula:
M = 75 * v^2 * r
What to do:
In a nearby active spiral galaxy (NGC 4258), 21 million light years away from Earth, there is an object called a maser—sort of a giant natural microwave laser—that can be observed orbiting something at the centre of the galaxy that is probably a black hole. The maser is 0.3 light years from the black hole (r) and is orbiting it at a speed of 900 Km/sec (v).
Calculate the size (i.e.: number of solar masses) of the black hole at the centre of this galaxy. (NOTE: you can look up more about this system in Nature #373, p. 103, 12 Jan 1996)
There is a giant galaxy called M87 in the centre of the Virgo cluster 65 million light years from Earth. Similar observations and calculations show a black hole at the centre of M87 with a mass of two billion suns!