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Chapter 7: Problem 38

An FM radio station broadcasts at \(99.5 \mathrm{MHz}\). Calculate the wavelength of the corresponding radio waves.

Short Answer

Expert verified
The wavelength of the radio waves for this FM radio station is approximately \(3.0151 \times 10^{-6}\) meters.

Step by step solution

01

Identify given values and the formula to be used.

The given frequency is \(99.5 \mathrm{MHz}\). Since 1 MHz is equal to \(1\times10^6 \mathrm{Hz}\), we can write the frequency as \(99.5 \times 10^6 \mathrm{Hz}\). The formula we need to use is: $$ v = f\lambda $$ where \(v\) is the speed of light, \(f\) is the frequency, and \(\lambda\) is the wavelength.
02

Convert units of the speed of light into meters per second.

The speed of light, \(v\), is approximately \(3\times10^8 \mathrm{m/s}\). Since the frequency is given in Hz and we want the wavelength in meters, we can use this value for the speed of light directly.
03

Rearrange the formula to solve for the wavelength.

To find the wavelength \(\lambda\), we can rearrange the formula as: $$ \lambda = \frac{v}{f} $$
04

Substitute the given values into the formula and solve for the wavelength.

Now we can plug in the values for the speed of light and the frequency: $$ \lambda = \frac{(3 \times 10^8 \mathrm{m/s})}{(99.5 \times 10^6 \mathrm{Hz})} $$
05

Calculate the wavelength and simplify.

Divide the values and simplify to find the wavelength: $$ \lambda = \frac{3 \times 10^8}{99.5 \times 10^6} \mathrm{m} $$ $$ \lambda \approx 3.0151 \times 10^6 \mathrm{m} $$ The wavelength of the radio waves for this FM radio station is approximately \(3.0151 \times 10^6\) meters.

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