Frequency f= nv/2l
for first resonance
325*2*0.254= v ---eqn 2
for second resonance:
325*2*0.774 = 2v ---eqn 1

subtracting eqn 1 and 2 we get:

325*2*(0.774-0.254) = v
therefore v= 338 m/s

Introduction:
In this question, we are given the frequency of a tuning fork and the lengths at which the first two resonances are observed. We need to calculate the speed of sound in air.

Resonance:
Resonance occurs when an object is forced to vibrate at its natural frequency by another vibrating object. In the case of a tuning fork, resonance occurs when the length of the air column in a tube matches a multiple of the wavelength of the sound produced by the tuning fork.

Calculating Wavelength:
To calculate the speed of sound, we first need to determine the wavelength of the sound produced by the tuning fork. We can use the formula:

\(\text{wavelength} = \frac{2 \times \text{length}}{n}\)

where length is the length of the air column and n is the number of the resonance observed.

Resonance 1:
Given length for the first resonance is 25.4 cm and n = 1. Substituting these values into the formula, we get:

\(\text{wavelength}_1 = \frac{2 \times 25.4}{1} = 50.8 \, \text{cm}\)

Resonance 2:
Given length for the second resonance is 77.4 cm and n = 2. Substituting these values into the formula, we get:

\(\text{wavelength}_2 = \frac{2 \times 77.4}{2} = 77.4 \, \text{cm}\)

Calculating Speed of Sound:
The speed of sound can be calculated using the formula:

\(\text{speed} = \text{frequency} \times \text{wavelength}\)

Substituting the values of frequency (325 Hz) and wavelength (50.8 cm) for the first resonance, we get:

\(\text{speed}_1 = 325 \times 50.8 = 16540 \, \text{cm/s}\)

Substituting the values of frequency (325 Hz) and wavelength (77.4 cm) for the second resonance, we get:

\(\text{speed}_2 = 325 \times 77.4 = 25155 \, \text{cm/s}\)

Average Speed of Sound:
To get a more accurate value for the speed of sound, we can take the average of the two speeds calculated:

\(\text{average speed} = \frac{\text{speed}_1 + \text{speed}_2}{2}\)

Substituting the values, we get:

\(\text{average speed} = \frac{16540 + 25155}{2} = 20847.5 \, \text{cm/s}\)

Conclusion:
The speed of sound in air is approximately 20847.5 cm/s.