Problem 61 - Algebra and Trigonometry with Analytic Geometry

Question

Refer to Exercises 67-68 in Section 3.3. If \(v_{0}\) is the wind speed at height \(h_{0}\) and if \(v_{1}\) is the wind speed at height \(h_{1}\), then the vertical wind shear can be described by the equation $$ \frac{v_{0}}{v_{1}}=\left(\frac{h_{0}}{h_{1}}\right)^{P} $$ where \(P\) is a constant. During a one-year period in Montreal, the maximum vertical wind shear occurred when the winds at the 200 -foot level were \(25 \mathrm{mi} / \mathrm{hr}\) while the winds at the 35 -foot level were \(6 \mathrm{mi} / \mathrm{hr}\). Find \(P\) for these conditions.

Step-by-Step Solution

Answer

The constant \( P \) is approximately 0.6443.

Step 1: Understand the given equation

The equation given to describe vertical wind shear is \( \frac{v_{0}}{v_{1}} = \left(\frac{h_{0}}{h_{1}}\right)^{P} \), where \( v_0 \) and \( v_1 \) are wind speeds at heights \( h_0 \) and \( h_1 \), respectively, and \( P \) is a constant.

Step 2: Assign given values to variables

Given that the wind speed at the 200-foot level is 25 mi/hr, we assign \( v_0 = 25 \) and \( h_0 = 200 \). Similarly, at the 35-foot level, the wind speed is 6 mi/hr, so \( v_1 = 6 \) and \( h_1 = 35 \).

Step 3: Substitute known values

Substitute \( v_0 = 25 \), \( v_1 = 6 \), \( h_0 = 200 \), and \( h_1 = 35 \) into the equation. This results in: \[ \frac{25}{6} = \left(\frac{200}{35}\right)^P \]

Step 4: Solve for the constant P

First, compute the left side and the inner fraction: - \( \frac{25}{6} = 4.1667 \)- \( \frac{200}{35} \approx 5.7143 \) Now, equate and solve for \( P \): \[ 4.1667 = (5.7143)^P \] Take the logarithm of both sides: \[ \log(4.1667) = P \cdot \log(5.7143) \]Solve for \( P \): \[ P = \frac{\log(4.1667)}{\log(5.7143)} \approx 0.6443 \]

Step 5: Verification

Verify the computed \( P \) by substituting it back into the original equation to check the equality holds. Use a calculator to ensure accuracy when taking powers and logarithms.

Key Concepts

Wind Speed

Wind speed refers to how fast the wind is moving, typically measured in miles per hour (mi/hr) or meters per second (m/s). In the context of vertical wind shear, wind speed is evaluated at different heights above the ground. For example, if the wind speed at a 200-foot height is measured at 25 mi/hr, it means the air particles at this elevation are traveling at this velocity. Similarly, a 6 mi/hr wind speed at a 35-foot height denotes how quickly the wind moves closer to the ground. Understanding wind speed at various heights is crucial for meteorological studies and applications, such as determining the stability of structures like buildings or wind turbines.

Height

Height in wind shear studies depicts the elevation above the ground surface where wind speed is measured. Precise measurement of height is crucial in analyzing vertical wind shear, as different layers of the atmosphere exhibit varying wind behavior. In our example, heights of 200 feet and 35 feet were used to measure respective wind speeds of 25 mi/hr and 6 mi/hr. These heights help meteorologists model the changes in wind patterns due to terrain features, atmospheric conditions, or changes in air pressure. Ultimately, knowing the wind speed at different heights allows for predicting weather patterns and understanding climate impacts.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-life situations. For vertical wind shear, the model is represented by the equation \( \frac{v_{0}}{v_{1}} = \left(\frac{h_{0}}{h_{1}}\right)^{P} \), where \( v_0 \) and \( v_1 \) represent wind speeds, \( h_0 \) and \( h_1 \) represent heights, and \( P \) is a constant. This equation models the relationship between wind speed changes at varying heights. Such models are beneficial in simulations to predict how winds might shift under different conditions. Mathematical equations provide a structured way to interpret data, simulate scenarios, and derive conclusions that may not be evident from raw observations alone.

Exponents

In mathematics, exponents are used to denote repeated multiplication of a number by itself. When discussing vertical wind shear, exponents help in defining how the ratio of heights affects the wind speed ratio. In the formula \( (\frac{h_{0}}{h_{1}})^{P} \), the exponent \( P \) determines the rate at which wind speed changes relative to changes in height. Understanding exponents is key because they provide the power to express complex relationships succinctly. For instance, calculating \( 5.7143^P \) gives us insights into the vertical wind shear phenomenon by showing how wind behaves differently at various altitudes, captured by the constant \( P \).