Final answer:
The length of the string is approximately 5.4 meters, which can be determined by analyzing the given standing wave function and considering that the string is in the 6th resonant mode.
Step-by-step explanation:
The student asked how to determine the length of the string in a resonance lab experiment with a given standing wave function y(x,t)=0.03sin(7x)cos(120t), where the string is in the 6th resonant mode. In a standing wave, the number of half-wavelengths (antinodes or loops) that fit into the length of the string corresponds to the mode of resonance. Since the student's string is in the 6th resonant mode, this means there are 6 half-wavelengths (λ/2) along the length of the string. The wave function given can be analyzed to find that 7x corresponds to the wave number k, where k=2π/λ. Therefore, we can solve for the wavelength (λ) with the equation 7 = 2π/λ, which gives us λ = 2π/7 meters. Since the 6th resonant mode has 6 half-wavelengths along the string, we find the length (L) of the string by multiplying the half-wavelength by 12 (as there are 2 half-wavelengths in a full wavelength, and we have 6 full wavelengths for the 6th mode). Thus, L = 6 * λ = 6 * (2π/7) = 12π/7 ≈ 5.4 meters, which is the length of the string.