To find the approximate depth of water for a tsunami traveling at 200 kilometers per hour using the equation [tex]\( S = 356 \sqrt{d} \)[/tex], we need to solve for [tex]\( d \)[/tex].
Here’s the step-by-step process to determine the depth:
1. Identify the given values and the equation:
- Given: [tex]\( S = 200 \)[/tex] km/h
- Equation: [tex]\( S = 356 \sqrt{d} \)[/tex]
2. Substitute the given speed [tex]\( S \)[/tex] into the equation:
[tex]\[ 200 = 356 \sqrt{d} \][/tex]
3. Isolate the square root term:
Divide both sides of the equation by 356:
[tex]\[ \sqrt{d} = \frac{200}{356} \][/tex]
4. Simplify the fraction:
[tex]\[ \sqrt{d} = \frac{200}{356} = \frac{100}{178} \approx 0.5618 \][/tex]
5. Square both sides to solve for [tex]\( d \)[/tex]:
[tex]\[ d = (0.5618)^2 \][/tex]
[tex]\[ d \approx 0.3156 \][/tex]
So, after following these steps, we find that the depth [tex]\( d \approx 0.316 \)[/tex] kilometers.
Now, we compare this result with the given options:
- [tex]\( 0.32 \)[/tex] km
- [tex]\( 0.75 \)[/tex] km
- [tex]\( 1.12 \)[/tex] km
- [tex]\( 3.17 \)[/tex] km
The closest value to [tex]\( 0.316 \)[/tex] km is [tex]\( 0.32 \)[/tex] km.
Therefore, the approximate depth of the water for a tsunami traveling at 200 kilometers per hour is:
[tex]\[ \boxed{0.32 \text{ km}} \][/tex]
