A college student is at a concert and really wants to hear the music, so she sits between two in-phase loudspeakers, which point toward each other and are 60.8 m apart. The speakers emit sound at a frequency of 428.75 Hz. At the midpoint between the speakers, there will be constructive interference, and the music will be at its loudest. At what distance closest to the midpoint (and along the line connecting the loudspeakers) could she also sit to experience the loudest sound? (Use 343.0 m/s for the speed of sound.)

To find the distance closest to the midpoint where the college student can also experience the loudest sound, we need to determine the locations of the points of constructive interference.

Constructive interference occurs when the path lengths from each loudspeaker to the point of interest differ by an integer multiple of the wavelength. The formula for the path length difference is given by:

ΔL = n * λ

Where:
ΔL is the path length difference
n is an integer representing the number of complete wavelengths between the loudspeakers
λ is the wavelength of the sound wave

The path length difference ΔL can be calculated by determining the difference in distances from each loudspeaker to the point of interest. The distance from each loudspeaker to the point is the distance between the loudspeakers (60.8 m) plus or minus the distance between the point and each loudspeaker.

The wavelength λ can be calculated using the speed of sound formula:

v = f * λ

Where:
v is the speed of sound (343.0 m/s)
f is the frequency of the sound wave (428.75 Hz)

First, let's calculate the wavelength:

λ = v / f
= 343.0 m/s / 428.75 Hz
≈ 0.8 m

Now, let's calculate the path length difference ΔL:

ΔL = n * λ

To find the distance closest to the midpoint where constructive interference occurs, we need to find the smallest positive value of n that satisfies the condition. We can do this by checking the path length differences for different values of n and finding the smallest one.

Let's start by calculating ΔL for n = 0, 1, -1, 2, -2, 3, -3, and so on:

ΔL(0) = 0 * λ = 0
ΔL(1) = 1 * λ = 1 * 0.8 m = 0.8 m
ΔL(-1) = -1 * λ = -1 * 0.8 m = -0.8 m
ΔL(2) = 2 * λ = 2 * 0.8 m = 1.6 m
ΔL(-2) = -2 * λ = -2 * 0.8 m = -1.6 m
ΔL(3) = 3 * λ = 3 * 0.8 m = 2.4 m
ΔL(-3) = -3 * λ = -3 * 0.8 m = -2.4 m

Based on the calculations, the smallest positive value of ΔL is 0.8 m. This means that the college student can sit at a distance of 0.8 m from the midpoint, towards either loudspeaker, to experience the loudest sound.

Therefore, the distance closest to the midpoint where the college student can also experience the loudest sound is approximately 60.8 m + 0.8 m = 61.6 m from either loudspeaker along the line connecting them.