To determine the frequency of an ultraviolet wave traveling through a vacuum given its wavelength, we can use the relationship between the speed of light [tex]\( c \)[/tex], frequency [tex]\( f \)[/tex], and wavelength [tex]\( \lambda \)[/tex]. The formula is:
[tex]\[ c = f \times \lambda \][/tex]
Where:
- [tex]\( c \)[/tex] is the speed of light, approximately [tex]\( 3.0 \times 10^8 \)[/tex] meters per second.
- [tex]\( \lambda \)[/tex] is the wavelength of the wave, given as [tex]\( 4.0 \times 10^{-7} \)[/tex] meters.
- [tex]\( f \)[/tex] is the frequency we need to find.
Rearranging the formula to solve for the frequency:
[tex]\[ f = \frac{c}{\lambda} \][/tex]
Substituting the given values:
[tex]\[ f = \frac{3.0 \times 10^8 \, \text{m/s}}{4.0 \times 10^{-7} \, \text{m}} \][/tex]
Performing the division:
First, calculate the coefficient:
[tex]\[ \frac{3.0}{4.0} = 0.75 \][/tex]
Then, handle the powers of ten separately:
[tex]\[ 10^8 \div 10^{-7} = 10^{8 - (-7)} = 10^{15} \][/tex]
Now combine the coefficient with the power of ten:
[tex]\[ f = 0.75 \times 10^{15} \][/tex]
However, to express the frequency in standard scientific notation with two significant figures, we adjust the coefficient to be between 1 and 10:
[tex]\[ 0.75 \times 10^{15} = 7.5 \times 10^{14} \][/tex]
Thus, the frequency of the ultraviolet wave, written in scientific notation to two significant figures, is:
[tex]\[ \boxed{7.5 \times 10^{14} \, \text{Hz}} \][/tex]
