A man stands at the midpoint between two speakers that are broadcasting an amplified static hiss uniformly in all directions. The speakers are 37.0 m apart and the total power of the sound coming from each speaker is 0.448 W. Find the total sound intensity the man hears when (a) he is at his initial position halfway between the speakers, and (b) after he has walked 6.0 m directly towards one of the speakers.

To find the total sound intensity the man hears, we can use the formula:

I = P / (4πr^2)

where I is the sound intensity, P is the power, and r is the distance from the source.

(a) When the man is initially at the midpoint between the two speakers, the distance from each speaker to the man is 37.0 m / 2 = 18.5 m.

Using the formula, the sound intensity from each speaker is:

I = 0.448 W / (4π(18.5 m)^2) = 0.00152 W/m^2

Since the sound from each speaker is uniformly in all directions, the total sound intensity the man hears is the sum of the intensities from each speaker:

Total sound intensity = 2 × 0.00152 W/m^2 = 0.00304 W/m^2

(b) After the man walks 6.0 m directly towards one of the speakers, his distance from that speaker is now (18.5 m - 6.0 m) = 12.5 m. The distance from the other speaker is still 18.5 m + 6.0 m = 24.5 m.

Using the formula, the sound intensity from the first speaker is:

I1 = 0.448 W / (4π(12.5 m)^2) = 0.00229 W/m^2

And the sound intensity from the second speaker is:

I2 = 0.448 W / (4π(24.5 m)^2) = 0.000677 W/m^2

The total sound intensity the man hears is the sum of the intensities from each speaker:

Total sound intensity = 0.00229 W/m^2 + 0.000677 W/m^2 = 0.00297 W/m^2

Therefore, the total sound intensity the man hears after walking 6.0 m directly towards one of the speakers is 0.00297 W/m^2.