Problem 8 - Calculus

Question

The wind speed, \(W,\) in meters per second, at a distance \(x\) km from the center of a hurricane is given by \(W=h(x).\) (a) Give the the units of \(d W / d x.\) (b) For a certain hurricane, \(h^{\prime}(15)>0 .\) What does this tell you about the hurricane?

Step-by-Step Solution

Answer

(a) The units are meters per second per kilometer. (b) The wind speed increases at 15 km from the center.

Step 1: Understand the Derivative

The derivative \( \frac{dW}{dx} \) represents the rate at which the wind speed changes with respect to the distance from the hurricane's center. This tells us how the wind speed varies as you move further or closer to the hurricane.

Step 2: Determine Units for \( \frac{dW}{dx} \)

Wind speed \( W \) is measured in meters per second \((\text{m/s})\), and distance \( x \) is in kilometers \((\text{km})\). So the units for \( \frac{dW}{dx} \) will be \( \frac{\text{m/s}}{\text{km}} \), i.e., meters per second per kilometer.

Step 3: Analyze the Meaning of \( h^{\prime}(15) > 0 \)

The condition \( h^{\prime}(15) > 0 \) means that at a distance of 15 km from the hurricane's center, the wind speed is increasing. This indicates that as the distance from the center increases, the wind speed is getting stronger at that particular point.

Key Concepts

Rate of Change

The rate of change is an important concept in calculus, especially when dealing with real-life scenarios such as wind speeds in hurricanes. When we talk about the rate of change, we refer to how one quantity changes as another quantity changes. In the context of our exercise, we're interested in how the wind speed, given by the function \( W = h(x) \), changes as the distance from the center of the hurricane, \( x \), changes.

This is precisely what the derivative \( \frac{dW}{dx} \) measures. It tells us the slope of the tangent line to the curve at any point \( x \). If the derivative is positive, like \( h^{\prime}(15) > 0 \), it means the wind speed is increasing as you move further away from the hurricane's center. If it's negative, the wind speed is decreasing. A zero derivative indicates no change in wind speed at that point. Recognizing this rate of change helps meteorologists understand hurricane dynamics more effectively. It shows the behavior of the storm's wind at various distances, crucial for predicting impacts.

Units of Measurement

Units of measurement play a significant role in accurately interpreting and communicating scientific data. In the exercise, the wind speed, \( W \), is measured in meters per second (\( \text{m/s} \)), reflecting how far wind particles travel over time. The distance from the hurricane's center, \( x \), is measured in kilometers (\( \text{km} \)).

When we calculate the derivative \( \frac{dW}{dx} \), the resulting units are a fraction composed of these two units. Specifically, the units will be meters per second per kilometer (\( \frac{\text{m/s}}{\text{km}} \)). This unit tells us how much the wind speed changes per every kilometer away from the hurricane's center. It's crucial to maintain these units throughout calculations because they provide context and allow comparisons. Without clear units, the data could be misinterpreted or compared incorrectly. Understanding units of measurement helps clarify the magnitude and significance of the rate of change in physical phenomena.

Wind Speed

Wind speed is a critical factor in weather phenomena and particularly in predicting and understanding hurricanes. It drills down into how fast air is moving and is often crucial for determining the potential damage a hurricane can cause. In our exercise, the wind speed around the hurricane follows a mathematical function \( W = h(x) \).

Wind speed, like in our exercise, is influenced by the distance from the hurricane's center. Changes in speed, described by the derivative, indicate how the storm may intensify or weaken. For instance, if \( h^{\prime}(15) > 0 \), it tells us that at 15 km from the center, wind speed is getting stronger. Understanding these variations is vital for safety and planning, as areas with increasing speeds could face greater risks.

Beyond the immediate application, wind speed data feeds into larger meteorological models. These models assist in forecast accuracy, helping predict the storm's path and potential landfall areas. Accurate knowledge of wind speed helps warn residents, guide evacuation plans, and optimize response activities. It's the insight derived from analyzing these speeds that can make a significant difference in catastrophe prevention and management.