Final answer:
The wavelength for the n = 6 mode of a vibrating string fixed at both ends and 1.0 meter long is 0.333 meters. The wavelength of the sound produced in air by the string would depend on the frequency of the vibration, which is not given.
Step-by-step explanation:
When a string is fixed at both ends and vibrated, it can form standing waves. These waves have specific wavelengths and frequencies that depend on the properties of the string and the mode of vibration. Considering the n = 6 mode, it means that there are 6 half wavelengths in the length of the string.
The length of the string (L) is given as 1.0 meter. For the n = 6 mode, this corresponds to 6 half wavelengths or 3 full wavelengths fitting into the length of the string. The wavelength (λ) of the standing wave can be found by dividing the length of the string by the number of half wavelengths:
λ = 2L / n
λ = 2 * 1.0 m / 6
λ = 0.333 m
For the sound wave in the air produced by the vibrating string, the wavelength of the sound wave is directly related to the speed of sound (Us) and the frequency of the vibration (f). The wavelength (λ_s) is the speed of sound divided by the frequency:
λ_s = Us / f
However, in the current context, additional information would be required to answer part (b) precisely as the frequency of oscillation for the n = 6 mode has not been provided.