Question

You observe an X-ray source to vary on a timescale of 0.001 seconds. What is the upper limit for the size of the X-ray emitting region? What is the mass of a black hole with a Schwarzschild radius of this size?

Answer 1

1) simply that the object must be of a size that light takes no more than 0.001 seconds to cross from one side of it to the other.

Light travels in 1 second 300,000 km.
In 0.001 seconds, light travels 300 km
And 300 km is the upper size limit for the object.

2) The size of a black hole, as determined by the radius of the event horizon, or Schwarzschild radius, is roughly proportional to the mass M through
radius = ~ 2.95 * M/Msun km

so a 300 km radius = ~ 2.95 * M
M = ~ 300 / 2.95
= ~ 101.7 Msun