Problem 87 - Algebra 2

Question

Weather Using data for the effect of temperature and wind on an exposed face, the National Weather Service uses the following formula. Wind Chill Index \(=35.74+0.6215 T-35.75 V^{0.16}+0.4275 T V^{0.16}\) \(T\) is the temperature in degrees Fahrenheit and \(V\) is the velocity of the wind in miles per hour. Frostbite occurs in about 15 minutes when the wind chill index is about \(-20 .\) Find the wind speed that produces a wind chill index of \(-20\) when the temperature is \(5^{\circ} \mathrm{F}\) .

Step-by-Step Solution

Answer

The wind speed that results in a wind chill index of \(-20\) when the temperature is \(5^{\circ} F\) can be found by solving the equation derived from the formula provided.

Step 1: Replace known values

First, replace the values of the Wind Chill Index and Temperature (\(T\)) into the given formula. The equation becomes: \(-20 = 35.74 + 0.6215 \times 5 - 35.75 V^{0.16} + 0.4275 \times 5 \times V^{0.16}\)

Step 2: Simplify the equation

Simplify the equation by performing multiplication and addition where possible. You get: \(-20 = 38.8525 - 35.75 V^{0.16} + 2.1375 \times V^{0.16}\)

Step 3: Combine like terms

Combine the terms involving \(V^{0.16}\) to make the equation simpler. This turns the equation into: \(-20 = 38.8525 - 33.6125 V^{0.16}\)

Step 4: Solve for \(V^{0.16}\)

Now, collect all terms involving \(V^{0.16}\) on one side and the remaining on the other side of the equality. The equation becomes: \(33.6125 V^{0.16} = 38.8525 + 20\)

Step 5: Simplify and solve for \(V\)

By performing the addition on the right side, and then dividing both sides by 33.6125, we solve for \(V^{0.16}\). After finding \(V^{0.16}\), we raise both sides of the equation to the power of \(\frac{1}{0.16}\) (which is the same as taking the sixth root) to find the value of \(V\). This will give us the wind speed that produces the wind chill index of \(-20\) when the temperature is \(5^{\circ} F\).

Key Concepts

Temperature Effects

Weather has a significant influence on how cold we feel. The air temperature, usually measured in degrees Fahrenheit or Celsius, affects our perception of cold. When the air temperature is low, our bodies lose heat faster, which can cause us to feel colder. But temperature alone isn't the only factor.
With temperatures at 5°F, it might already feel quite cold, but when combined with wind, the cold effect intensifies. As such, understanding the effect of temperature on perceived coldness is crucial, as it underscores the importance of dressing warmly and taking precautions during colder months.

• Air temperature directly impacts how quickly our bodies lose heat.
• Lower temperatures increase the risks of cold-related health problems.

Wind Speed Calculation

Wind speed plays a crucial role in determining how cold we feel, due to its ability to speed up heat loss from our bodies. When air moves past our skin, it removes heat that our bodies generate to keep warm. The faster the wind blows, the quicker this heat is carried away, leading to a feeling of increased cold.
In the wind chill formula, wind speed is represented as $V$ in miles per hour (mph). Calculating the correct wind speed helps in predicting how cold it will feel, aiding in making preparations for outdoor activities.

• Wind speed affects how quickly body heat is lost.
• Higher wind speeds make temperatures feel colder than they are.

Frostbite Risk

Frostbite is a serious medical condition that occurs when skin tissues freeze due to extreme cold conditions. The risk of frostbite increases significantly with a low wind chill index. The wind chill index provides a better sense of when frostbite might occur than air temperature alone because it accounts for the cooling effects of both wind and low temperature.
For instance, a wind chill index of $-20$ indicates a high frostbite risk, with exposed skin potentially freezing in minutes.

• Frostbite can occur quickly when the wind chill is low.
• Protection against wind and cold is essential to prevent frostbite.

Mathematical Modeling

Mathematical models, like the wind chill formula, are tools we use to predict and quantify complex systems. This formula combines multiple variables (temperature and wind speed) to estimate the perceived coldness. Such models help turn real-world phenomena into computable predictions.
Solving these models involves understanding and manipulating the equation to find unknown values, such as the wind speed required to reach a given wind chill index. By plugging known values into the formula, we apply algebraic manipulation to arrive at the desired result.

• Models simplify complex weather phenomena into equations.
• They help predict weather effects, aiding in preparations.

Weather Formulas

Weather formulas, like the wind chill index, provide essential insights into how weather conditions affect us. These formulas are derived from empirical data and involve variables like temperature and wind speed to deliver practical information.
For instance, the wind chill index formula helps scale raw data into understandable and actionable information, guiding clothing choices and outdoor activity planning. It is an example of how mathematics and science work together to interpret and respond to nature.

• Weather formulas convert measurements into understandable indexes.
• They are invaluable for planning and adapting to weather conditions.