Problem 76 - Brief Applied Calculus

Question

GENERAL: Tsunamis The speed of a tsunami (popularly known as a tidal wave, although it has nothing whatever to do with tides) depends on the depth of the water through which it is traveling. At a depth of \(d\) feet, the speed of a tsunami will be \(s(d)=3.86 \sqrt{d}\) miles per hour. Find the speed of a tsunami in the Pacific basin where the average depth is 15,000 feet.

Step-by-Step Solution

Answer

The speed of the tsunami is approximately 472.74 miles per hour.

Step 1: Understand the formula

The formula provided for the speed of a tsunami is \( s(d) = 3.86 \sqrt{d} \). This indicates that the speed depends on the square root of the depth, multiplied by 3.86.

Step 2: Substitute the average depth into the equation

Substitute the average depth \( d = 15,000 \) feet into the formula: \[ s(15,000) = 3.86 \sqrt{15,000} \].

Step 3: Calculate the square root of the depth

Calculate \( \sqrt{15,000} \), which is approximate to 122.47.

Step 4: Multiply by the constant factor

Now, multiply the result of the square root by 3.86: \( s(15,000) = 3.86 \times 122.47 \).

Step 5: Compute the final speed

Perform the multiplication: \( 3.86 \times 122.47 \approx 472.74 \). Hence, the speed of the tsunami at a depth of 15,000 feet is approximately 472.74 miles per hour.

Key Concepts

Tsunami Speed Calculation

A tsunami, often mistakenly referred to as a tidal wave, is a powerful natural phenomenon caused by undersea disturbances such as earthquakes or volcanic eruptions. Their speed in the ocean is crucial in determining how quickly they will reach distant coastlines. The speed of a tsunami is calculated based on the depth of the water it is traveling through. This relationship is given by the formula:

• \( s(d) = 3.86 \sqrt{d} \)

Here, \( s(d) \) represents the speed in miles per hour, and \( d \) is the depth of the water in feet. Understanding this equation helps scientists predict how fast a tsunami will move through the ocean.

To find the speed of a tsunami in a particular location, such as the Pacific basin with an average depth of 15,000 feet, we substitute the value of \( d = 15,000 \) into the formula. The speed calculation becomes a straightforward task of mathematical substitution and multiplication.

Square Root Function

The square root function is a key component in the calculation of tsunami speed. It is a mathematical operation where a number is multiplied by itself to achieve a specific value. In the context of our formula, \( \sqrt{d} \) is the core element used to calculate speed, where \( d \) is the depth of the water.

For example, to find the square root of 15,000, we solve the equation \( \sqrt{15,000} \approx 122.47 \). This result is approximate because square roots often result in irrational numbers, which cannot be expressed exactly using simple fractions or decimals.

Recognizing and effectively calculating square roots is essential in understanding many other scientific and engineering concepts beyond tsunami modeling. It allows for the transformation of quadratic relationships into linear approximations, facilitating easier and more intuitive predictions and computations.

Mathematical Modeling

Mathematical modeling involves using mathematical structures and equations to represent real-world phenomena. It plays a critical role in the natural sciences, engineering, and economics. In the case of tsunamis, the formula \( s(d) = 3.86 \sqrt{d} \) is a perfect example of modeling where mathematics helps explain and predict the behavior of these massive ocean waves.

By understanding and utilizing such a model, scientists can forecast the potential impact and speed of tsunamis, aiding in better preparedness and response strategies worldwide. Models make complex systems more comprehensible by reducing them to essential components and predictable patterns.

Mathematical modeling is not just about computations; it's about transforming abstract concepts into practical tools that can achieve real-world applications. It requires a thorough understanding of both the underlying mathematics and the physical processes at play, ensuring that predictions are scientifically sound and practically useful.