Problem 59 - College Algebra

Question

In a hurricane, the wind pressure varies directly as the square of the wind velocity. If wind pressure is a measure of a hurricanes destructive capacity, what happens to this destructive power when the wind speed doubles?

Step-by-Step Solution

Answer

When the wind speed doubles, the destructive power of the hurricane quadruples.

Step 1: Understand the Relationship

Firstly, it's important to remember the mathematical relationship that governs direct variation, which is \(y = kx^m\), where \(y\) is the dependent variable, \(x\) is the independent variable, \(k\) is the constant of variation and \(m\) is the exponent. Here, the wind pressure is the dependent variable (\(y\)), wind velocity is the independent variable (\(x\)), and the exponent (\(m\)) is 2 since the pressure varies directly as the square of the velocity.

Step 2: Calculate the Effect of Doubling Velocity

To find the destructive power when the wind speed doubles, let's replace \(x\) with \(2x\) in the equation \(y = kx^2\), which gives \((y = k(2x)^2)\), this simplifies to \(y = 4kx^2\). This means, if the wind velocity doubles, the wind pressure will also increase by four times, provided the constant (\(k\)) remains unchanged.

Key Concepts

Wind Pressure

Wind pressure is the force exerted by the wind per unit area. It is a crucial factor in determining the impact a hurricane can have on surroundings. The mathematical relationship for wind pressure relative to wind velocity is expressed as a direct variation. This indicates that wind pressure increases as the square of the wind velocity increases.

In mathematical terms, if the wind velocity is represented as variable \(x\), and the wind pressure as \(y\), the equation is given by \(y = kx^2\). Where \(k\) is the constant of variation. This equation shows that small increases in wind speed can significantly increase wind pressure, and thereby the potential damage a hurricane might cause.

• The formula helps in predicting hurricane impacts.
• Understanding wind pressure assists in designing structures to withstand harsh weather conditions.

Wind Velocity

Wind velocity refers to the speed at which wind moves over the Earth's surface. In the context of hurricanes, wind velocity is a critical factor in determining the storm's severity and potential for devastation. Wind velocity is an independent variable in the wind pressure equation.

As wind velocity increases, the pressure exerted by the wind increases exponentially, given the direct variation relationship. When wind velocity doubles, the pressure quadruples, showing a highly non-linear relationship.

Understanding this concept is crucial because:

• It highlights why small differences in wind speed can lead to much larger differences in potential damage.
• It underscores the importance of accurate wind speed measurements during hurricane forecasts and monitoring.

Hurricane Destructive Capacity

A hurricane's destructive capacity is largely measured by its wind pressure, which directly correlates with wind velocity. The destructive capacity is a measure of how much damage a hurricane can inflict on structures, vegetation, and coastlines. By understanding the direct variation relationship between wind pressure and velocity, we can better estimate a hurricane's potential damage.

For instance, when wind speed doubles, the destructive power is not doubled but increased by four times. This emphasizes the destructive potential of strong hurricanes.

• The concept is crucial for emergency preparedness and response.
• It plays a vital role in shaping building codes and hurricane-resistant designs.

Understanding hurricane destructive capacity helps individuals, communities, and authorities prepare for potential impacts effectively, emphasizing the need for accurate weather predictions and timely evacuations when necessary.