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The speed that a tsunami can travel is modeled by the equation [tex]S=356 \sqrt{d}[/tex], where [tex]S[/tex] is the speed in kilometers per hour and [tex]d[/tex] is the average depth of the water in kilometers.

What is the approximate depth of water for a tsunami traveling at 200 kilometers per hour?

A. [tex]0.32 \, \text{km}[/tex]
B. [tex]0.75 \, \text{km}[/tex]
C. [tex]1.12 \, \text{km}[/tex]
D. [tex]3.17 \, \text{km}[/tex]

Sagot :

GinnyAnswer
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]
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