Question

1. an fm radio station broadcasts at a frequency of 89.5 mhz. calculate the wavelength of this broadcast in meters. (1 mhz = 10⁶ hz) a. 3.35 m b. 2.98 m c. 2.69 m d. 5.93 m e. 3.64 m

Explanation:

Step1: Recall the wave equation

The relationship between the speed of light \( c \), frequency \( f \), and wavelength \( \lambda \) is given by \( c = \lambda f \), where \( c = 3.00\times10^{8}\, \text{m/s} \) (speed of light in a vacuum), \( f \) is the frequency, and \( \lambda \) is the wavelength. We need to solve for \( \lambda \), so we rearrange the formula to \( \lambda=\frac{c}{f} \).

Step2: Convert the frequency to Hz

The frequency \( f = 89.5\, \text{MHz} \). Since \( 1\, \text{MHz}=10^{6}\, \text{Hz} \), we convert the frequency:
\( f = 89.5\times 10^{6}\, \text{Hz}=8.95\times 10^{7}\, \text{Hz} \)

Step3: Calculate the wavelength

Substitute \( c = 3.00\times 10^{8}\, \text{m/s} \) and \( f = 8.95\times 10^{7}\, \text{Hz} \) into the formula \( \lambda=\frac{c}{f} \):
\( \lambda=\frac{3.00\times 10^{8}\, \text{m/s}}{8.95\times 10^{7}\, \text{Hz}} \)
First, divide the coefficients: \( \frac{3.00}{8.95}\approx0.335 \)
Then, divide the powers of 10: \( \frac{10^{8}}{10^{7}} = 10 \)
Multiply the results: \( 0.335\times 10=3.35\, \text{m} \)

Answer:

A. 3.35 m