Problem 32
Question
Because wind speed enhances the loss of heat from the skin, we feel colder when there is wind than when there is not. The wind chill temperature is what the temperature would have to be with no wind in order to give the same chilling effect. The wind chill temperature, \(W\), is given by \(\mathrm{W}(v, T)=91.4-\frac{(10.45+6.68 \sqrt{\mathrm{v}}-0.447 \mathrm{v})(457-5 \mathrm{~T})}{110}\)where \(T\) is the temperature measured by a thermometer, in degrees Fahrenheit, and \(v\) is the speed of the wind, in miles per hour. Find the wind chill temperature in each case. Round to the nearest degree. $$ T=20^{\circ} \mathrm{F}, v=20 \mathrm{mph} $$
Step-by-Step Solution
Verified Answer
The wind chill temperature is -10°F.
1Step 1: Identify the variables
From the formula, \( W(v, T) = 91.4 - \frac{(10.45 + 6.68 \sqrt{v} - 0.447 v)(457 - 5 T)}{110} \), identify \( T = 20 \) degrees Fahrenheit and \( v = 20 \) mph.
2Step 2: Compute square root and product terms
Calculate \( \sqrt{v} \) for \( v = 20 \). This gives us \( \sqrt{20} \approx 4.47 \). Use this to find \( 6.68 \sqrt{20} \approx 6.68 \times 4.47 \approx 29.87 \).
3Step 3: Calculate the expression inside the parentheses
Substitute and compute: \( 10.45 + 29.87 - 0.447 \times 20 \). First, calculate \( 0.447 \times 20 = 8.94 \). Now add: \( 10.45 + 29.87 - 8.94 = 31.38 \).
4Step 4: Evaluate the remaining expression
Find \( 457 - 5 \times 20 = 457 - 100 = 357 \). Multiply by the expression from the previous step: \( 31.38 \times 357 = 11200.46 \).
5Step 5: Compute the fraction
Divide the result from the previous step by 110: \( \frac{11200.46}{110} = 101.82 \).
6Step 6: Calculate the wind chill temperature
Subtract the result from 91.4: \( W = 91.4 - 101.82 = -10.42 \). Round \(-10.42\) to the nearest degree to get \(-10\).
Key Concepts
Wind Speed Effect on TemperatureTemperature MeasurementChilling Effect Formula
Wind Speed Effect on Temperature
The way wind affects our perception of temperature is fascinating. Wind can make the air temperature feel much colder than it actually is. This happens because wind increases the rate at which heat is removed from our skin.
Imagine standing outside on a calm day when the temperature is 20°F. Now, imagine a day with the same temperature but with strong wind blowing. Despite the thermometer reading staying the same, the windy day will feel much colder.
This is because the wind helps carry away the warm layer of air that surrounds our body and our skin loses heat faster. It's important to be aware of wind conditions, especially during colder months, as they can significantly impact how cold we feel.
Imagine standing outside on a calm day when the temperature is 20°F. Now, imagine a day with the same temperature but with strong wind blowing. Despite the thermometer reading staying the same, the windy day will feel much colder.
This is because the wind helps carry away the warm layer of air that surrounds our body and our skin loses heat faster. It's important to be aware of wind conditions, especially during colder months, as they can significantly impact how cold we feel.
Temperature Measurement
Understanding temperature measurement is key to discussing any aspect involving temperature, such as wind chill. Thermometers measure the thermal energy of the environment, typically in degrees Fahrenheit, Celsius, or Kelvin. In wind chill calculations, we use Fahrenheit.
When we measure temperature with a thermometer, it provides us with the true air temperature, which doesn’t account for external factors like wind. While the given temperature reading is still accurate in terms of the environment's thermal energy, it might not reflect how our bodies perceive this temperature in windy conditions.
Therefore, while thermometers will tell you the nominal temperature, actual experienced warmth or coldness should always consider additional factors such as wind speed, which can enhance the chilling effect on our bodies.
When we measure temperature with a thermometer, it provides us with the true air temperature, which doesn’t account for external factors like wind. While the given temperature reading is still accurate in terms of the environment's thermal energy, it might not reflect how our bodies perceive this temperature in windy conditions.
Therefore, while thermometers will tell you the nominal temperature, actual experienced warmth or coldness should always consider additional factors such as wind speed, which can enhance the chilling effect on our bodies.
Chilling Effect Formula
The chilling effect formula is a mathematical model used to estimate the wind chill temperature—how cold it feels when the wind is blowing. This is expressed through the formula:
\[W(v, T) = 91.4 - \frac{(10.45 + 6.68 \sqrt{v} - 0.447 v)(457 - 5 T)}{110}\]
Here:
This approach allows meteorologists to provide a more accurate description of how temperature truly feels with the wind factored in, ensuring that people are properly informed and can dress appropriately for the weather conditions.
\[W(v, T) = 91.4 - \frac{(10.45 + 6.68 \sqrt{v} - 0.447 v)(457 - 5 T)}{110}\]
Here:
- \( W \) is the wind chill temperature.
- \( T \) is the actual air temperature in degrees Fahrenheit.
- \( v \) is the wind speed in miles per hour.
This approach allows meteorologists to provide a more accurate description of how temperature truly feels with the wind factored in, ensuring that people are properly informed and can dress appropriately for the weather conditions.
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