Two small forward-facing speakers are 2.50 m apart.They are both emitting,in phase with each other,a sound of frequency 1100 Hz in a room where the speed of sound is 344 m/s.A woman is standing opposite the midpoint between the speakers and is initially 35.0 m from the midpoint.As she slowly walks parallel to the line connecting the speakers,at what angle θ (relative to the centerline coming outward from the midpoint between the speakers)will she first hear no sound?
A) 0.063°
B) 3.6°
C) 7.2°
D) 1.8°
E) 11°

Question

Quizplus · Accepted Answer

The first position where the woman hears no sound is when there is a path difference of half a wavelength ($\lambda/2$) between the sound waves reaching her from the two speakers. The wavelength ($\lambda$) of the sound can be calculated using the formula $\lambda = v/f$, where $v$ is the speed of sound (344 m/s) and $f$ is the frequency (1100 Hz). This gives $\lambda = 344 / 1100 = 0.3127$ m. Therefore, $\lambda/2 = 0.1564$ m. The path difference corresponds to the difference in distances from each speaker to the woman's position, forming a right triangle with the line connecting the speakers and the line from the midpoint to the woman's position. Using the sine rule, $\sin(\theta) = \frac{\lambda/2}{d}$, where $d = 2.50$ m is the distance between the speakers. Substituting the values, $\sin(\theta) = \frac{0.1564}{2.50}$, giving $\theta = \arcsin(0.06256)$, which is approximately $3.6^\circ$.

Two Small Forward-Facing Speakers Are 2