To solve this problem, we want to determine the depth of water [tex]\( d \)[/tex] that corresponds to a tsunami traveling at a speed of [tex]\( S = 200 \)[/tex] kilometers per hour using the given formula:
[tex]\[ S = 356 \sqrt{d} \][/tex]
We will evaluate each given depth and see which one provides a speed closest to 200 kilometers per hour.
1. Depth = 0.32 kilometers
Let's compute the speed:
[tex]\[
S = 356 \sqrt{0.32}
\][/tex]
Simplifying, we get:
[tex]\[
S \approx 356 \times 0.5657 \approx 201.38 \text{ km/h}
\][/tex]
This is quite close to 200 km/h.
2. Depth = 0.75 kilometers
Let's compute the speed:
[tex]\[
S = 356 \sqrt{0.75}
\][/tex]
Simplifying, we get:
[tex]\[
S \approx 356 \times 0.866 \approx 308.31 \text{ km/h}
\][/tex]
This speed is significantly higher than 200 km/h.
3. Depth = 1.12 kilometers
Let's compute the speed:
[tex]\[
S = 356 \sqrt{1.12}
\][/tex]
Simplifying, we get:
[tex]\[
S \approx 356 \times 1.058 \approx 376.75 \text{ km/h}
\][/tex]
This speed is also higher than 200 km/h.
4. Depth = 3.17 kilometers
Let's compute the speed:
[tex]\[
S = 356 \sqrt{3.17}
\][/tex]
Simplifying, we get:
[tex]\[
S \approx 356 \times 1.78 \approx 633.84 \text{ km/h}
\][/tex]
This speed is much higher than 200 km/h.
Comparing all these speeds, the depth that gives a speed closest to 200 km/h is:
Depth = 0.32 kilometers, which corresponds to a speed of approximately 201.38 km/h.
Therefore, the approximate depth of water for a tsunami traveling at 200 kilometers per hour is 0.32 kilometers.
