Problem 51 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A formula used by meteorologists to calculate the wind chill temperature (the temperature that you feel in still air that is the same as the actual temperature when the presence of wind is taken into consideration) is\(T=f(t, s)=35.74+0.6215 t-35.75 s^{0.16}+0.4275 t s^{0.16}$$(s \geq 1)\) where \(t\) is the actual air temperature in degrees Fahrenheit and \(s\) is the wind speed in mph. a. What is the wind chill temperature when the actual air temperature is \(32^{\circ} \mathrm{F}\) and the wind speed is \(20 \mathrm{mph}\) ? b. If the temperature is \(32^{\circ} \mathrm{F}\), by how much approximately will the wind chill temperature change if the wind speed increases from \(20 \mathrm{mph}\) to \(21 \mathrm{mph}\) ?

Step-by-Step Solution

Answer

The wind chill temperature when the actual air temperature is 32°F and the wind speed is 20 mph is approximately 29.95°F. When the wind speed increases from 20 mph to 21 mph, keeping the actual air temperature constant at 32°F, the wind chill temperature decreases by approximately 1.72°F.

Step 1: a. Wind Chill Temperature Calculation

First, we will use the formula to calculate the wind chill temperature when t=32°C and s=20 mph: \(T = f(t, s) = 35.74 + 0.6215t - 35.75s^{0.16} + 0.4275ts^{0.16}\) Plug in the values of t=32°F and s=20 mph into the formula: \(T = f(32, 20) = 35.74 + 0.6215(32) - 35.75(20)^{0.16} + 0.4275(32)(20)^{0.16}\) Now, calculate the equation: \(T ≈ 35.74 + 19.888 - 35.75(2.639) + 0.4275(32)(2.639)\) \(T ≈ 35.74 + 19.888 - 94.394 + 35.912\) \(T ≈ 29.955\) So, the wind chill temperature when the actual air temperature is 32°F and the wind speed is 20 mph is approximately 29.95°F.

Step 2: b. Wind Chill Temperature Change

The problem asks how much the wind chill temperature will change if the wind speed increases from 20 mph to 21 mph while the temperature remains the same (32°F). First, calculate the wind chill temperature when s=21 mph using the same formula: \(T = f(32, 21) = 35.74 + 0.6215(32) - 35.75(21)^{0.16} + 0.4275(32)(21)^{0.16}\) Now, calculate the equation: \(T ≈ 35.74 + 19.888 - 35.75(2.698) + 0.4275(32)(2.698)\) \(T ≈ 35.74 + 19.888 - 96.359 + 36.964\) \(T ≈ 28.233\) So, the wind chill temperature when the actual air temperature is 32°F and the wind speed is 21 mph is approximately 28.23°F. Now, find the difference in the wind chill temperatures when the wind speed increased from 20 mph to 21 mph: ΔT = T(21 mph) - T(20 mph) ≈ 28.233 - 29.955 ≈ -1.722 The wind chill temperature decreases by approximately 1.72°F when the wind speed increases from 20 mph to 21 mph, keeping the actual air temperature constant at 32°F.

Key Concepts

Meteorology

Meteorology is the science dedicated to understanding the atmosphere and how its processes affect our environment. It plays a crucial role in weather forecasting, allowing us to predict changes in weather patterns. One important aspect that meteorologists consider is the wind chill effect, which refers to how cold the air feels to humans due to the combined effect of low temperatures and wind. The wind can remove heat from your body, making it feel colder than the actual air temperature. This is why meteorologists have developed formulas to calculate wind chill, such as the one used in the original exercise. Knowing the wind chill temperature helps people make informed decisions about outdoor activities and prepare appropriately for severe weather conditions.

Temperature Formulas

Temperature formulas are mathematical expressions that help us calculate specific temperature conditions based on various variables. For example, the wind chill formula in our exercise, \( T=f(t, s)=35.74+0.6215t-35.75s^{0.16}+0.4275ts^{0.16} \), allows us to compute the perceived coldness felt when wind speeds get involved.In this equation:

• \( T \) is the wind chill temperature
• \( t \) is the actual air temperature in degrees Fahrenheit
• \( s \) is the wind speed in miles per hour (mph)

This formula takes into account both the actual air temperature and the wind speed to provide a more accurate picture of how cold the weather feels. Calculating wind chill accurately is not just about taking a simple average; it involves understanding the interaction between temperature and wind speed. By substituting the specific values of temperature and wind speed into the formula, you can determine how cold it will feel at any given time.

Mathematical Problem Solving

Mathematical problem solving involves using logical reasoning and mathematical principles to address specific questions and find solutions. In the context of our wind chill exercise, we utilized this approach to calculate perceived temperatures under varying wind speeds. First, we tackled part (a) by substituting the given values for air temperature (32°F) and wind speed (20 mph) into the wind chill formula to determine the perceived temperature. This step-by-step plugging of values is a key part of problem solving. Next, for part (b), we focused on understanding how a change in one variable (wind speed increasing from 20 mph to 21 mph) affects the wind chill temperature. By recalculating using the updated wind speed, we were able to discern that the wind chill temperature decreased, emphasizing how small changes in input can lead to different outcomes. Such exercises improve understanding of how interrelated factors affect temperature perception and foster critical thinking skills, which are essential in broader meteorological and mathematical contexts.