Sound with frequency 1220 Hz leaves a room through a doorway with a width of 1.06 m . At what minimum angle relative to the centerline perpendicular to the doorway will someone outside the room hear no sound

θ =0.133 radians = 7.62°

Explanation:

f= 1220 Hz, d=1.06 m

Solution: As no sound is heard so destructive interference

θ = (m + 1/2) λ / d   (here m =0)

so find λ= v/f   ( v=speed of sound = 344 m/s)

λ = 344 m/s /1220 Hz

λ = 0.282 m

Now θ = ( 0 + 0.5 ) × 0.282 m /1.06 m

θ = 0.133 radians

180 degree = π radians

1 radian = 180 degree / π

θ = 7.62°

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west