To determine the approximate depth of water when a tsunami is traveling at a speed of 200 kilometers per hour, we can use the given formula for the speed of a tsunami:
[tex]\[ S = 356 \sqrt{d} \][/tex]
where:
- [tex]\( S \)[/tex] is the speed of the tsunami in kilometers per hour (km/h),
- [tex]\( d \)[/tex] is the average depth of the water in kilometers (km).
Given:
[tex]\[ S = 200 \, \text{km/h} \][/tex]
We need to solve for [tex]\( d \)[/tex].
### Steps to Solve:
1. Start from the given equation:
[tex]\[ 200 = 356 \sqrt{d} \][/tex]
2. Isolate [tex]\( \sqrt{d} \)[/tex] by dividing both sides of the equation by 356:
[tex]\[ \sqrt{d} = \frac{200}{356} \][/tex]
3. Calculate the value of the fraction:
[tex]\[ \frac{200}{356} \approx 0.5618 \][/tex]
4. Now, solve for [tex]\( d \)[/tex] by squaring both sides:
[tex]\[ d = (0.5618)^2 \][/tex]
5. Calculate the square of 0.5618:
[tex]\[ d \approx 0.3156 \][/tex]
Thus, the approximate depth of the water when a tsunami is traveling at 200 kilometers per hour is approximately:
[tex]\[ d \approx 0.3156 \, \text{km} \][/tex]
### Comparing with the Given Choices:
- 0.32 km
- 0.75 km
- 1.12 km
- 3.17 km
The closest value to our calculated result [tex]\( 0.3156 \)[/tex] km is [tex]\( 0.32 \)[/tex] km.
Therefore, the approximate depth of water for a tsunami traveling at 200 kilometers per hour is:
[tex]\[ \boxed{0.32 \, \text{km}} \][/tex]
