To determine the speed of sound in air at a temperature of [tex]\( 113^\circ \text{F} \)[/tex], we need to follow a step-by-step process that involves converting temperatures and using known relationships for the speed of sound. Here's the detailed solution:
1. Convert temperature from Fahrenheit to Celsius:
[tex]\[
\text{Temperature in }^\circ \text{C} = (\text{Temperature in }^\circ \text{F} - 32) \times \frac{5}{9}
\][/tex]
Plugging in [tex]\( 113^\circ \text{F} \)[/tex]:
[tex]\[
113^\circ \text{F} - 32 = 81
\][/tex]
Then,
[tex]\[
81 \times \frac{5}{9} = 45^\circ \text{C}
\][/tex]
So, the temperature in Celsius is [tex]\( 45^\circ \text{C} \)[/tex].
2. Convert the temperature from Celsius to Kelvin:
The relationship between Celsius and Kelvin is:
[tex]\[
\text{Temperature in } K = \text{Temperature in }^\circ \text{C} + 273
\][/tex]
For [tex]\( 45^\circ \text{C} \)[/tex]:
[tex]\[
45 + 273 = 318 \text{ K}
\][/tex]
Therefore, the temperature in Kelvin is 318 K.
3. Calculate the speed of sound at the given temperature:
The speed of sound in air increases with temperature. At [tex]\( 0^\circ \text{C} \)[/tex], the speed of sound is [tex]\( 331 \text{ m/s} \)[/tex]. The speed of sound [tex]\( v \)[/tex] at temperature [tex]\( T \)[/tex] in Kelvin compared to its speed at [tex]\( T_0 \)[/tex] in Kelvin (standard temperature, which is 273 K) can be found using the formula:
[tex]\[
v = v_0 \times \left( \frac{T}{T_0} \right)^{0.5}
\][/tex]
where [tex]\( v_0 \)[/tex] is the speed of sound at [tex]\( 0^\circ \text{C} \)[/tex] or 273 K. Given [tex]\( v_0 = 331 \text{ m/s} \)[/tex], [tex]\( T = 318 \text{ K} \)[/tex], and [tex]\( T_0 = 273 \text{ K} \)[/tex]:
[tex]\[
v = 331 \times \left( \frac{318}{273} \right)^{0.5}
\][/tex]
[tex]\[
\frac{318}{273} \approx 1.165
\][/tex]
[tex]\[
\left( 1.165 \right)^{0.5} \approx 1.079
\][/tex]
[tex]\[
v = 331 \times 1.079 \approx 357.24 \text{ m/s}
\][/tex]
Therefore, the speed of sound in air at a temperature of [tex]\( 113^\circ \text{F} \)[/tex] is approximately [tex]\( 357.24 \text{ m/s} \)[/tex].
