Approximately how long would it take a telegraph signal to cross the united states from the east coast to the west coast? (telegraph signals travel at about the speed of light.)

If the coast-to-coast distance is about 2900 miles, it will take about ...

... (2900 mi)/(186000 mi/s) ≈ 15.6 ms

for the signal to travel coast-to-coast at light speed.

There are time delays due to the amplifiers and repeaters required in the path, and due to the fact that the signals don't actually travel at the speed of light in a wire.

A test of internet speed between Seattle and Washington, DC shows a round-trip time of about 87 ms when traffic is light. Modern communication channels tend to be faster than the telegraph (especially when measured in words per minute).