An ambulance with a siren emitting a whine at 1800 hz overtakes and passes a cyclist pedaling a bike at 2.23 m/s. after being passed, the cyclist hears a frequency of 1790 hz. how fast is the ambulance moving? (take the speed of sound in air to be 343 m/s.

It's the Doppler Effect that "up and down sound." 

I was on Yahoo--- doesn't have an option for Credientials or Site credit, so I'll just put this in the quotes:

The formula for doppler effect is always (s is speed and f is frequency): 
f_perceived.by.observer = f_of.emitted.wave * (s_wave + s_observer) / (s_wave + s_source.of.wave) 

And you should pay attention to the signs: 
s_observer is positive if the receiver is moving towards the source, negative otherwise 
s_source.of.wave is positive if the source is moving away from the observer, negative otherwise 

Applying it to this case: 

s_source.of.wave = ? (positive), speed of ambulance 
s_observer = + 2.44 m/s speed cyclist 
f_of.emitted.wave =1800 Hz frequency of whine 
f_perceived.by.observer = 1760 frequency heard by cyclist 
s_wave = 343 m/s speed of sound in air 

Now you know every value in the equation for doppler effect except by s_source.of.wave, so you can solve for s_source.of.wave.