Two speakers S_1 and S_2 , powered by a common amplifier, are...

The distance between the two speakers $S_1$ and $S_2$ is $d = 1.0 - (-1.0) = 2.0 \text{ m}$.

The wavelength of the sound wave is given by:

$\lambda = \dfrac{v}{f} = \dfrac{330}{600} = 0.55 \text{ m}$

For a point at a very large distance from the origin at an angle $\theta$ with the X-axis, the path difference between the sound waves reaching the man from $S_1$ and $S_2$ is:

$\Delta x = d \sin \theta$

The man starts from the X-axis (\theta = 0^{}^{{}^{\circ}}), where there is a central maximum. As he moves parallel to the Y-axis, he will experience the first maximum of sound intensity when the path difference equals one wavelength ($\lambda$).

$\Delta x = \lambda$

$d \sin \theta = \lambda$

Substituting the values, we get:

$2.0 \sin \theta = 0.55$

$\sin \theta = \dfrac{0.55}{2.0} = 0.275$

We know that \sin 15^{}^{{}^{\circ}} = \dfrac{\sqrt{3}-1}{2\sqrt{2}} \approx 0.259. Since $0.275$ is slightly greater than $0.259$, the angle $\theta$ must be slightly greater than 15^{}^{{}^{\circ}}.

From the given options, the closest angle is 16^{}^{{}^{\circ}}.