two loudspeakers are set to produce waves at 255Hz and 258Hz respectively. a tuning fork of unknown frequency is sounded together with each speaker in turn. with the 255Hz speaker it produces 10 beats in 10second and with the 258Hz speaker it produces 10beats in 5second. what is the frequency of the tuning fork?

10/10 = 1 beat/s

10/5 = 2 beats/s

255 + 1 = 256
258 - 2 = 256

so 256

Some historical trivia: 256 Hz used to be middle C, but with equal-tempered tuning based upon A = 440 Hz, middle C has been changed to 262 Hz.

To find the frequency of the tuning fork, we need to understand how beats are formed when two waves of slightly different frequencies interfere with each other.

When two waves with slightly different frequencies are played simultaneously, they interfere constructively and destructively, resulting in a phenomenon called beats. Beats are variations in the loudness or amplitude of the combined wave that we can hear. The number of beats per second is equal to the difference in frequencies between the two waves.

In this case, the tuning fork is sounded with two different speakers, one producing a wave at 255Hz and the other producing a wave at 258Hz.

Let's first calculate the beat frequency when the tuning fork is sounded with the 255Hz speaker. According to the information given, there are 10 beats that occur in 10 seconds. Therefore, the beat frequency can be calculated as follows:

Beat frequency = (Number of beats) / (Time taken)
= 10 beats / 10 seconds
= 1 beat per second

Next, let's calculate the beat frequency when the tuning fork is sounded with the 258Hz speaker. There are 10 beats occurring in 5 seconds, so we can calculate the beat frequency as follows:

Beat frequency = (Number of beats) / (Time taken)
= 10 beats / 5 seconds
= 2 beats per second

Now, we have two values for the beat frequency with each speaker: 1 beat per second and 2 beats per second. The tuning fork's frequency is the average of these two beat frequencies.

Tuning fork frequency = (Frequency of first speaker + Frequency of second speaker) / 2
= (255Hz + 258Hz) / 2
= 513Hz / 2
= 256.5Hz

Therefore, the frequency of the tuning fork is approximately 256.5Hz.