Problem 33 - Calculus and its applications

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. $$ T=20^{\circ} \mathrm{F}, v=40 \mathrm{mph} $$

Step-by-Step Solution

Answer

The wind chill temperature is \(-22^{\circ} F\).

Step 1: Substitute Values into the Formula

Start by substituting the given values of temperature \( T = 20 \) and wind speed \( v = 40 \) into the wind chill temperature formula: \[W(40, 20) = 91.4 - \frac{(10.45 + 6.68 \sqrt{40} - 0.447 \cdot 40)(457 - 5 \cdot 20)}{110}\]

Step 2: Calculate the Square Root of Wind Speed

Find the square root of the wind speed, which is \( \sqrt{40} \). You can approximate this value as \( \sqrt{40} \approx 6.32 \).

Step 3: Simplify the Expression Inside the Parentheses

Substitute the approximation into the equation to simplify:\[10.45 + 6.68 \times 6.32 - 0.447 \times 40\]Calculate each part:- \( 6.68 \times 6.32 = 42.21 \)- \( 0.447 \times 40 = 17.88 \)Add and subtract these values:\[10.45 + 42.21 - 17.88 = 34.78\]

Step 4: Calculate the Expression for Temperature

Substitute \( T = 20 \) into the expression \( 457 - 5T \):\[457 - 5 \times 20 = 457 - 100 = 357\]

Step 5: Substitute and Simplify the Remaining Expression

Substitute the simplified expressions into the formula:\[W(40, 20) = 91.4 - \frac{34.78 \times 357}{110}\]Calculate \( 34.78 \times 357 \):\[34.78 \times 357 = 12418.86\]Then, divide by 110:\[\frac{12418.86}{110} \approx 112.90\]

Step 6: Final Calculation of Wind Chill Temperature

Subtract the value calculated above from 91.4:\[W(40, 20) = 91.4 - 112.90 = -21.5\]Round \(-21.5\) to the nearest degree to get the final wind chill temperature: \[ -22^{\circ} \]

Key Concepts

Heat Loss

Heat loss from the body is a natural process, and it's the reason we feel cold when exposed to lower temperatures. Wind can have a significant impact on how we perceive temperature because it accelerates the rate at which heat is lost from our skin.
The body normally loses heat through several mechanisms:

• Radiation: Emission of heat from our body into the surrounding environment.
• Convection: Heat transfer through air currents, which is especially prevalent on windy days.
• Conduction: Direct transfer of heat through contact with colder surfaces or air.

Wind plays a crucial role by increasing convection. It moves the warm air surrounding our body and replacing it with colder air, thereby increasing the rate at which heat is lost.
This enhanced heat loss is what makes us perceive the temperature as being colder than it actually is, leading to the concept of wind chill temperature. When wind speed is factored in, a more realistic perception of the cold can be assessed, specified as wind chill temperature.

Wind Speed

Wind speed is a key factor that influences the rate of perceived temperature and contributes to the wind chill effect. It is typically measured in miles per hour (mph) or kilometers per hour (km/h).
Higher wind speeds drastically increase the rate of heat loss through convection, making temperatures feel colder on the skin than the actual air temperature.

• The faster the wind, the more rapid the displacement of warm air from the skin's surface.
• Increased wind speed corresponds to a higher effect of cooling, leading to a lower wind chill temperature.

Understanding wind speed's effect on heat loss can be critical in assessing risks in cold weather. For instance, high winds combined with low temperatures require more robust insulation and wind barriers in clothing to prevent hypothermia, which occurs when the body loses heat faster than it can produce it.
Accurately measuring wind speed is crucial for calculating the wind chill temperature, providing valuable information for outdoor planning and safety.

Temperature Measurement

Temperature measurement is crucial when assessing the conditions for wind chill, as it provides a baseline for calculations. Temperature is usually measured in degrees Fahrenheit in the United States or degrees Celsius in most other parts of the world.
The measurement of air temperature gives us a direct view of how hot or cold the environment actually is, without factoring in wind effects. Thermometers are the common instrument used to measure ambient temperature.

• A mercury or alcohol thermometer is often used in traditional settings.
• In digital format, thermocouples offer high precision and can be connected to weather sensors for more comprehensive data.

While static temperature measurements tell how cold or warm it is, they don't account for the body's perception affected by wind or humidity. Distinguishing actual temperature from felt temperature like wind chill allows for more effective decision-making regarding clothing, heating, and outdoor activities.
Accurate measurement of temperature provides part of the equation necessary for determining the wind chill temperature, combining it with wind speed to evaluate the potential impact on humans exposed to cold conditions.