Question

violet light has a frequency of ( 7.26\times10^{14}) hz and travels at a speed of ( 3.00\times10^{8}) m/s. the wavelength of violet light, rounded to the nearest nanometer, is (\text{<image>}) nm.

Explanation:

Step1: Recall the wave - speed formula

The formula for the speed of a wave is \(v = f\lambda\), where \(v\) is the speed of the wave, \(f\) is the frequency, and \(\lambda\) is the wavelength. We need to solve for \(\lambda\), so \(\lambda=\frac{v}{f}\).

Step2: Substitute the given values

Given \(v = 3.00\times10^{8}\text{ m/s}\) and \(f = 7.26\times 10^{14}\text{ Hz}\). Then \(\lambda=\frac{3.00\times 10^{8}}{7.26\times 10^{14}}\text{ m}\).
Using the rule of exponents \(\frac{a^{m}}{a^{n}}=a^{m - n}\), we have \(\lambda=\frac{3.00}{7.26}\times10^{8-14}\text{ m}\).
\(\frac{3.00}{7.26}\approx0.413\) and \(10^{8 - 14}=10^{- 6}\text{ m}\). So \(\lambda = 0.413\times10^{-6}\text{ m}\).

Step3: Convert meters to nanometers

Since \(1\text{ m}=10^{9}\text{ nm}\), then \(\lambda=0.413\times10^{-6}\times10^{9}\text{ nm}\).
Using the rule of exponents \(a^{m}\times a^{n}=a^{m + n}\), we get \(\lambda=0.413\times10^{3}\text{ nm}=413\text{ nm}\).

Answer:

$413$ nm