Final answer:
To determine the temperature of the air through which a sound wave is traveling, the speed of the sound wave must be calculated using its frequency and wavelength and then compared to the known speed of sound at 0°C. By solving for the temperature difference using the speed of sound in air and its known increase per degree Celsius, we find that the closest estimated temperature is 300 K.
Step-by-step explanation:
To find the temperature of the air that a sound wave is traveling through, one can use the relationship between the speed of sound, wavelength, and frequency of the wave, and relate that to temperature.
The speed of a sound wave (v) is given by the formula:
v = frequency (f) × wavelength (λ)
The frequency (f) can be found by taking the reciprocal of the period (T):
f = 1/T
Given a period (T) of 0.000313s, we can find the frequency:
f = 1 / 0.000313s ≈ 3195 Hz
Using the frequency and the wavelength (λ) of 0.11 m:
v = f × λ ≈ 3195 Hz × 0.11 m ≈ 351.45 m/s
We know that sound travels at 331 m/s in air at 0°C and that it increases with temperature. Since the speed we've found is above 331 m/s and it's known that at 20.0°C the speed of sound is 343 m/s, we estimate that the temperature will be somewhere between 0°C and 20°C. To find the exact temperature, we can use the formula that relates the speed of sound in air at different temperatures:
v = 331 m/s + (0.6 m/s/°C × temperature difference from 0°C)
This gives us: 351.45 = 331 + (0.6 × temperature difference from 0°C)
Solving for the temperature difference yields:
temperature difference = (351.45 - 331) / 0.6 ≈ 34.08°C
Adding this to 0°C to find the actual temperature:
Temperature = 0°C + 34.08°C = 34.08°C
Converting this to Kelvin:
Temperature in Kelvin = 34.08°C + 273.15 = 307.23 K
So, the closest answer to 307.23 K from the given options would be:
C) 300 K