To determine the speed of light in water, we can use the index of refraction for water and the known speed of light in a vacuum. The index of refraction ([tex]\(n\)[/tex]) of a material is defined as the ratio of the speed of light in a vacuum ([tex]\(c\)[/tex]) to the speed of light in the material ([tex]\(v\)[/tex]):
[tex]\[ n = \frac{c}{v} \][/tex]
Rearranging this formula to solve for the speed of light in the material, we get:
[tex]\[ v = \frac{c}{n} \][/tex]
Given:
- The speed of light in a vacuum ([tex]\(c\)[/tex]) is [tex]\( 3.0 \times 10^8 \, \text{m/s} \)[/tex].
- The index of refraction for water ([tex]\(n\)[/tex]) is 1.33.
Plugging these values into the formula, we calculate the speed of light in water:
[tex]\[ v = \frac{3.0 \times 10^8 \, \text{m/s}}{1.33} \][/tex]
Performing the division:
[tex]\[ v \approx 225563909.7744361 \, \text{m/s} \][/tex]
To express this result in terms of [tex]\( 10^8 \, \text{m/s} \)[/tex], we divide by [tex]\( 10^8 \)[/tex]:
[tex]\[ v \approx 2.255639097744361 \times 10^8 \, \text{m/s} \][/tex]
Now, rounding this to the nearest tenth:
[tex]\[ v \approx 2.3 \times 10^8 \, \text{m/s} \][/tex]
Thus, the speed of light in water, rounded to the nearest tenth, is:
[tex]\[ 2.3 \times 10^8 \, \text{m/s} \][/tex]
