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. 0.32 km
B. 0.75 km
C. 1.12 km
D. 3.17 km
Answer (2)
The approximate depth of water for a tsunami traveling at 200 kilometers per hour is 0.32 kilometers. This was calculated using the speed formula S = 356 d and isolating for depth. The answer is option A: 0.32 km.
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Substitute the given speed S = 200 into the equation S = 356 d .
Isolate the square root term: d = 356 200 .
Square both sides to solve for d : d = ( 356 200 ) 2 .
Calculate and approximate the depth: d ≈ 0.32 km. The final answer is 0.32 km .
Explanation
Understanding the Problem We are given the equation S = 356 d , where S is the speed of the tsunami in kilometers per hour and d is the average depth of the water in kilometers. We are also given that the tsunami is traveling at 200 kilometers per hour, so S = 200 . We want to find the approximate depth of the water, d .
Substituting the Value of S Substitute S = 200 into the equation S = 356 d :
200 = 356 d
Isolating the Square Root Term Divide both sides of the equation by 356 to isolate the square root term: d = 356 200
Solving for d Square both sides of the equation to solve for d :
d = ( 356 200 ) 2
Calculating d Calculate the value of d :
d = ( 356 200 ) 2 ≈ 0.3156
Approximating d Approximate the value of d to two decimal places: d ≈ 0.32
Final Answer The approximate depth of the water is 0.32 kilometers.
Examples
Understanding the speed of tsunamis based on water depth is crucial for early warning systems. For instance, if a tsunami is detected traveling at 200 km/h, knowing the relationship S = 356 d allows us to estimate the ocean depth in that area. This helps predict the tsunami's arrival time and potential impact on coastal regions, enabling timely evacuations and minimizing damage. This mathematical model plays a vital role in disaster preparedness and mitigation efforts.
