Question

a certain radio wave has a wavelength of 8×10^2 m, and a certain gamma ray has a wavelength of 2×10^(-12) m. which of the following is true?

a. the wavelength of the radio wave is forty billion times the wavelength of the gamma ray

b. the wavelength of the gamma ray is forty billion times the wavelength of the radio wave

c. the wavelength of the gamma ray is four billion times the wavelength of the radio wave

d. the wavelength of the radio wave is four billion times the wavelength of the gamma ray

Explanation:

Step1: Calculate the ratio of the radio - wave wavelength to the gamma - ray wavelength.

Let $\lambda_{r}$ be the wavelength of the radio wave and $\lambda_{g}$ be the wavelength of the gamma ray. $\lambda_{r}=8\times10^{2}\text{ m}$ and $\lambda_{g}=2\times10^{- 12}\text{ m}$. The ratio $\frac{\lambda_{r}}{\lambda_{g}}=\frac{8\times10^{2}}{2\times10^{-12}}$.

Step2: Use the rule of exponents $\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}$.

$\frac{8\times10^{2}}{2\times10^{-12}}=\frac{8}{2}\times10^{2-(-12)} = 4\times10^{14}$. Since $10^{9}$ is one billion, $4\times10^{14}=4\times10^{5}\times10^{9}$, which means 40 billion.

Answer:

A. The wavelength of the radio wave is forty billion times the wavelength of the gamma ray