To solve these questions, we will use the formula that relates the speed of light $c$, wavelength $\lambda$, and frequency $f$:

$c = \lambda \cdot f$

Where:

• $c$ is the speed of light, approximately $3.00 \times 10^8 \, \text{m/s}$.
• $\lambda$ is the wavelength in meters.
• $f$ is the frequency in Hertz (Hz).

Part 1: Calculate the Wavelength

Given:

• Frequency $f = 7.23 \times 10^{13} \, \text{Hz}$.
• We need to find $\lambda$ in nanometers (nm).

Steps:

1. Rearrange the formula to solve for $\lambda$: $\lambda = \frac{c}{f}$

2. Substitute $c = 3.00 \times 10^8 \, \text{m/s}$ and $f = 7.23 \times 10^{13} \, \text{Hz}$: $\lambda = \frac{3.00 \times 10^8}{7.23 \times 10^{13}}$

3. Calculate $\lambda$ in meters, then convert it to nanometers:

   • $1 \, \text{m} = 10^9 \, \text{nm}$.

Part 2: Calculate the Frequency

Given:

• Wavelength $\lambda = 3.74 \times 10^2 \, \text{nm}$.
• We need to find $f$ in Hertz (Hz).

Steps:

1. Convert $\lambda$ from nanometers to meters: $\lambda = 3.74 \times 10^2 \, \text{nm} = 3.74 \times 10^{-7} \, \text{m}$

2. Rearrange the formula to solve for $f$: $f = \frac{c}{\lambda}$

3. Substitute $c = 3.00 \times 10^8 \, \text{m/s}$ and $\lambda = 3.74 \times 10^{-7} \, \text{m}$: $f = \frac{3.00 \times 10^8}{3.74 \times 10^{-7}}$

4. Calculate $f$ in Hz.

Let me know if you would like the final numeric answers with calculations.

Would you like to go over each step in more detail or have any questions?

Additional Related Questions:

1. How would the wavelength change if the frequency increased?
2. How is wavelength related to the energy of a photon?
3. What are some applications of frequency and wavelength calculations in science?
4. Can this formula be applied to sound waves as well? Why or why not?
5. What is the significance of using the speed of light constant in these calculations?

Tip: Always make sure to convert units properly when working with frequency and wavelength calculations.