To determine the wavelength of X-rays given their frequency, we will use the inverse variation equation described:
[tex]\[ y = \frac{3 \times 10^8}{x} \][/tex]
where [tex]\( y \)[/tex] is the frequency in hertz (Hz) and [tex]\( x \)[/tex] is the wavelength in meters (m).
Given:
[tex]\[ y = 3 \times 10^{18} \text{ Hz} \][/tex]
We need to find the corresponding wavelength [tex]\( x \)[/tex]. Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{3 \times 10^8}{y} \][/tex]
Substitute the given frequency into the equation:
[tex]\[ x = \frac{3 \times 10^8}{3 \times 10^{18}} \][/tex]
Simplify the equation by canceling out the common factors:
[tex]\[ x = \frac{3 \times 10^8}{3 \times 10^{18}} = \frac{10^8}{10^{18}} \][/tex]
When dividing exponents with the same base, you subtract the exponents:
[tex]\[ x = 10^{8 - 18} = 10^{-10} \][/tex]
Thus, the calculated wavelength for X-rays with a frequency of [tex]\( 3 \times 10^{18} \text{ Hz} \)[/tex] is [tex]\( 1 \times 10^{-10} \text{ meters} \)[/tex].
Let's compare this result with the provided choices:
- [tex]\( 1 \times 10^{-10} \text{ m} \)[/tex]
- [tex]\( 3 \times 10^{-10} \text{ m} \)[/tex]
- [tex]\( 3 \times 10^{26} \text{ m} \)[/tex]
- [tex]\( 9 \times 10^{26} \text{ m} \)[/tex]
The calculated wavelength [tex]\( 1 \times 10^{-10} \text{ m} \)[/tex] matches exactly with the first choice.
Therefore, the wavelength for X-rays with a frequency of [tex]\( 3 \times 10^{18} \text{ Hz} \)[/tex] is:
[tex]\[ \boxed{1 \times 10^{-10} \text{ m}} \][/tex]
