To find the wavelength of violet light given its frequency and the speed of light, we will use the relationship between speed, frequency, and wavelength. This relationship is expressed by the formula:
[tex]\[ \text{wavelength} (\lambda) = \frac{\text{speed of light} (c)}{\text{frequency} (f)} \][/tex]
Given data:
- Frequency [tex]\( f = 7.26 \times 10^{14} \)[/tex] Hz
- Speed of light [tex]\( c = 3.00 \times 10^8 \)[/tex] m/s
Let's proceed with the step-by-step calculation:
1. Calculate the wavelength in meters:
[tex]\[ \lambda = \frac{c}{f} = \frac{3.00 \times 10^8 \text{ m/s}}{7.26 \times 10^{14} \text{ Hz}} \][/tex]
[tex]\[ \lambda = 4.1322314049586775 \times 10^{-7} \text{ meters} \][/tex]
2. Convert the wavelength from meters to nanometers:
Since [tex]\(1 \text{ meter} = 1 \times 10^9 \text{ nanometers}\)[/tex], we multiply:
[tex]\[ \lambda_{\text{nm}} = 4.1322314049586775 \times 10^{-7} \text{ meters} \times 10^9 = 413.22314049586777 \text{ nanometers} \][/tex]
3. Round the wavelength to the nearest nanometer:
[tex]\[ \lambda_{\text{nm}} \approx 413 \text{ nanometers} \][/tex]
Therefore, the wavelength of violet light, rounded to the nearest nanometer, is [tex]\( \boxed{413} \)[/tex] nm.
