Certainly! Let's go through the problem step-by-step.
We are given the following details:
- The observed frequency when the car is driving away, [tex]\( f = 240 \)[/tex] Hz.
- The stationary frequency, [tex]\( f_0 = 255 \)[/tex] Hz.
- The speed of sound, [tex]\( c = 761 \)[/tex] miles per hour.
We are to find the speed of the car [tex]\( v \)[/tex] using the Doppler effect formula for a source moving away from the observer:
[tex]\[ f = \left(\frac{c}{c + v}\right) f_0 \][/tex]
To solve for [tex]\( v \)[/tex], we first rearrange the equation:
1. Multiply both sides by [tex]\( (c + v) \)[/tex] to clear the fraction:
[tex]\[ f \cdot (c + v) = c \cdot f_0 \][/tex]
2. Distribute [tex]\( f \)[/tex] on the left side:
[tex]\[ f \cdot c + f \cdot v = c \cdot f_0 \][/tex]
3. Isolate the term involving [tex]\( v \)[/tex]:
[tex]\[ f \cdot v = c \cdot f_0 - f \cdot c \][/tex]
4. Factor [tex]\( c \)[/tex] out of the right side:
[tex]\[ f \cdot v = c \cdot (f_0 - f) \][/tex]
5. Finally, solve for [tex]\( v \)[/tex]:
[tex]\[ v = \frac{c \cdot (f_0 - f)}{f} \][/tex]
Now substitute the given values into the formula:
[tex]\[ v = \frac{761 \cdot (255 - 240)}{240} \][/tex]
Calculate the difference in frequencies:
[tex]\[ 255 - 240 = 15 \][/tex]
Multiply this difference by the speed of sound:
[tex]\[ 761 \cdot 15 = 11415 \][/tex]
Divide by the observed frequency:
[tex]\[ v = \frac{11415}{240} \][/tex]
[tex]\[ v = 47.5625 \][/tex]
Round to one decimal place:
[tex]\[ v = 47.6 \][/tex]
Therefore, the estimated speed of the car is [tex]\( 47.6 \)[/tex] miles per hour.
