In musical acoustics, a frequency ratio of 2: 1 is called an octave. Humans with extremely good hearing can hear sounds ranging from \(20 \mathrm{Hz}\) to \(20 \mathrm{kHz}\), which is approximately 10 octaves (since $2^{10}=1024 \approx 1000$ ). (a) Approximately how many octaves of visible light are humans able to perceive? (b) Approximately how many octaves wide is the microwave region?

First, let's define the limits of the visible light spectrum. The lower limit corresponds to the color red with a wavelength of \(700 \mathrm{nm}\), and the upper limit corresponds to the color violet with a wavelength of \(400 \mathrm{nm}\). We need to find the frequency at these wavelengths. Using the formula for speed of light, we can find the frequencies for lower and upper limits: Lower limit frequency: \(f_1 = \frac{v}{λ} = \frac{3 * 10^8 \mathrm{m/s}}{700 * 10^{-9} \mathrm{m}} \approx 4.29 * 10^{14} \mathrm{Hz}\) Upper limit frequency: \(f_2 = \frac{v}{λ} = \frac{3 * 10^8 \mathrm{m/s}}{400 * 10^{-9} \mathrm{m}} \approx 7.50 * 10^{14} \mathrm{Hz}\) Now we can find the frequency ratio between these limits: \(\frac{f_2}{f_1} \approx \frac{7.50 * 10^{14} \mathrm{Hz}}{4.29 * 10^{14} \mathrm{Hz}} \approx 1.75\) Next, we can find the number of octaves by taking the base-2 logarithm of the frequency ratio: Number of octaves of visible light: \(\log_2{(1.75)} \approx 0.81\) So, humans are able to perceive approximately \(0.81\) octaves of visible light.

The microwave region of the electromagnetic spectrum has a range of wavelengths approximately between \(1 \mathrm{mm}\) and \(100 \mathrm{cm}\). We will find the frequencies for these limits and determine the number of octaves. Lower limit frequency: \(f_1 = \frac{v}{λ} = \frac{3 * 10^8 \mathrm{m/s}}{100 * 10^{-2} \mathrm{m}} = 3 * 10^{9} \mathrm{Hz}\) Upper limit frequency: \(f_2 = \frac{v}{λ} = \frac{3 * 10^8 \mathrm{m/s}}{1 * 10^{-3} \mathrm{m}} = 3 * 10^{11} \mathrm{Hz}\) We can then find the frequency ratio between these limits: \(\frac{f_2}{f_1} = \frac{3 * 10^{11} \mathrm{Hz}}{3 * 10^{9} \mathrm{Hz}} = 100\) And finally, we can find the number of octaves by taking the base-2 logarithm of the frequency ratio: Number of octaves of microwave region: \(\log_2{(100)} \approx 6.64\) So, the microwave region is approximately \(6.64\) octaves wide.