Let's solve the problem step-by-step:
1. Calculate the absolute refractive index of glass and water:
The absolute refractive index of glass is given by [tex]\(\frac{8}{2}\)[/tex].
[tex]\[
n_{\text{glass}} = \frac{8}{2} = 4.0
\][/tex]
The absolute refractive index of water is given by [tex]\(\frac{4}{3}\)[/tex].
[tex]\[
n_{\text{water}} = \frac{4}{3} \approx 1.3333
\][/tex]
2. Determine the speed of light in glass:
The speed of light in glass is given as [tex]\(2 \times 10^8 \, \text{m/s}\)[/tex].
3. Find the speed of light in a vacuum (denoted as [tex]\(c\)[/tex]):
The refractive index is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium.
For glass, we have:
[tex]\[
n_{\text{glass}} = \frac{c}{v_{\text{glass}}}
\][/tex]
Rearranging for [tex]\(c\)[/tex]:
[tex]\[
c = n_{\text{glass}} \times v_{\text{glass}}
\][/tex]
Substituting in the given values:
[tex]\[
c = 4.0 \times 2 \times 10^8 \, \text{m/s} = 8 \times 10^8 \, \text{m/s}
\][/tex]
4. Calculate the speed of light in water:
Using the refractive index formula for water:
[tex]\[
n_{\text{water}} = \frac{c}{v_{\text{water}}}
\][/tex]
Rearranging for [tex]\(v_{\text{water}}\)[/tex]:
[tex]\[
v_{\text{water}} = \frac{c}{n_{\text{water}}}
\][/tex]
Substituting in the values:
[tex]\[
v_{\text{water}} = \frac{8 \times 10^8 \, \text{m/s}}{\frac{4}{3}}
\][/tex]
Simplifying this, we get:
[tex]\[
v_{\text{water}} = \frac{8 \times 10^8 \, \text{m/s}}{1.3333} \approx 6 \times 10^8 \, \text{m/s}
\][/tex]
Therefore, the speed of light in water is approximately [tex]\(6 \times 10^8 \, \text{m/s}\)[/tex].
None of the given options exactly matches [tex]\(6 \times 10^8 \, \text{m/s}\)[/tex], so it appears there may be an issue with the provided choices or the exact representation of values. However, based on the detailed calculations, we have determined that:
[tex]\[
v_{\text{water}} = 6 \times 10^8 \, \text{m/s}
\][/tex]
