Answer
(a) The person wearing headphones will hear the sound first, 0.30 seconds after the ball hits the bat.
(b) The person who hears the sound through the air hears the sound 0.10 seconds after the person wearing headphones hears the sound.
Work Step by Step
(a) The speed of sound at this temperature is $343~m/s$.
We can find the time $t_a$ for the person to hear the sound through the air:
$t_a = \frac{d}{v} = \frac{140~m}{343~m/s} = 0.40~s$
We can find the time $t_1$ for the sound to reach the microphone:
$t_1 = \frac{d}{v} = \frac{17~m}{343~m/s} = 0.050~s$
We can find the time $t_2$ for the signal to travel 75,000 km:
$t_2 = \frac{d}{c} = \frac{7.5\times 10^7~m}{3.0\times 10^8~m/s} = 0.25~s$
We can find the total time for the sound to reach the person wearing headphones:
$t = t_1+t_2 = (0.050~s)+(0.25~s) = 0.30~s$
The person wearing headphones will hear the sound first, 0.30 seconds after the ball hits the bat.
(b) The person who hears the sound through the air hears the sound 0.40 seconds after the ball hits the bat, which is 0.10 seconds after the person wearing headphones hears the sound.
