A quarter-wave monopole radio antenna (also called a Marconi antenna) consists of a long conductor of one quarter the length of the transmitted wavelength. The lower end of the antenna is connected to a conducting ground plane (often simply the Earth) which reflects the transmitted signal, creating an effective antenna length of a half wavelength

(a) What must be the height of the antenna (in m) for a radio station broadcasting at 604 kHz?
(b) What must be the height of the antenna (in m) for radio stations broadcasting at 1,710 kHz?

a) Height of the antenna (in m) for a radio station broadcasting at 604 kHz = 124.17 m

b)Height of the antenna (in m) for radio stations broadcasting at 1,710 kHz =43.86 m

Explanation:

(a) Radiowave wavelength= λ = c/f

As we know, Radiowave speed in the air = c = 3 x 10^8 m/s

f = frequency = 604 kHz = 604 x 10^3 Hz

Hence, wavelength = (3x10^8/604x10^3) m

λ = 496.69 m

So the height of the antenna BROADCASTING AT 604 kHz =  λ /4 = (496.69/4) m

= 124.17 m

(b) As we know , f = 1710 kHz = 1710 x 10^3 Hz  (1kHZ = 1000 Hz)

Hence, wavelength =  λ = (3 x 10^8/1710 x 10^3) m

 λ= 175.44 m

So, height of the antenna =  λ /4 = (175.44/4) m

= 43.86 m  

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