help please with part B Young\'s double-slit experiment underlies the instrument

Question

help please with part B

Young's double-slit experiment underlies the instrument landing system used to guide aircraft to safe landings when the visibility is poor. Although real systems are more complicated than the example described here, they operate on the same principles. A pilot is trying to align her plane with a runway, as suggested in Figure P37.11a. Two radio antennas A_1 and A_2 are positioned adjacent to the runway, separated by d = 37.0 m. The antennas broadcast unmodulated coherent radio waves at 31.0 MHz. Find the wavelength of the radio signal. The pilot "locks onto" the strong signal radiated along an interference maximum and steers the plane to keep the received signal strong. If she has found the central maximum, the plane will have just the right heading to land when it reaches the runway. Suppose instead the plane is flying along the first side maximum (Fig. P37.11b). How far to the side of the runway centerline will the plane be when it is 2.00 km from the antennas?

Explanation / Answer

here,

Distance from antennas, L = 2km = 2000 m
distance between the antennas, d = 41 m
frequency, f = 34Mhz = 34000000 Hz

Part a :
lamda = speed of sound / frequency
lamda = (3*10^8) / (34000000)
lamda = 8.8235 m

Part B:
The fist side maximum is at angle given by :
dSinA = n*Lamda
SinA = (1*8.8235/41)
SinA = 0.2152
A = arcSin(0.2152)
A = 12.4273 degrees

also
y = L*SinA
y = (2*1000)*Sin(12.4273)
y = 430.401 m

The plane will be 430.401 m from centerline

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