Problem 34 - 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. $$ T=-10^{\circ} \mathrm{F}, v=30 \mathrm{mph} $$

Step-by-Step Solution

Answer

The wind chill temperature is approximately -38°F.

Step 1: Write Down the Wind Chill Formula

The wind chill temperature \( W(v, T) \) is given by the formula: $$ W(v, T) = 91.4 - \frac{(10.45 + 6.68 \sqrt{v} - 0.447 v)(457 - 5T)}{110} $$ where \( T \) is the temperature in degrees Fahrenheit, and \( v \) is the wind speed in miles per hour.

Step 2: Substitute Known Values

Given \( T = -10 \) and \( v = 30 \), substitute these values into the formula: $$ W(30, -10) = 91.4 - \frac{(10.45 + 6.68 \sqrt{30} - 0.447 \times 30)(457 - 5 \times (-10))}{110} $$

Step 3: Simplify Inside the Bracket

Calculate the values inside the bracket of the formula: - Evaluate \( 6.68 \sqrt{30} \). - Replace \( v \) with 30 and calculate: \( 10.45 + 6.68 \times \sqrt{30} - 0.447 \times 30 \).

Step 4: Calculate the Inner Product

Calculate the product of the result from Step 3 and \( 457 - 5 \times (-10) \). - \( 457 - 5 \) multiplied by \( -10 \) gives \( 457 + 50 \). - Calculate the product.

Step 5: Divide by 110

Divide the result from Step 4 by 110. This will give the value to subtract from 91.4 in the formula.

Step 6: Final Calculation

Subtract the result from Step 5 from 91.4 to find the wind chill temperature. Round to the nearest integer to get the final answer.

Key Concepts

Heat Loss

When we talk about heat loss, we're usually referring to the way our body loses heat to the environment. This is a crucial aspect of how we perceive temperature. On cold and windy days, this is especially important.
Wind plays a significant role here. That's because moving air carries away heat from our body more quickly than still air. This increased heat loss from the skin is what makes us feel colder when it's windy.

• Heat moves away from the body when in contact with colder air.
• Wind increases the rate at which this heat moves, creating the chilling effect.
• This is because the wind increases the rate of heat transfer from the skin.

By understanding how wind affects heat loss, we're better able to comprehend why wind chill feels colder than the actual air temperature indicates.

Skin Cooling Effect

The skin cooling effect refers to how wind impacts the temperature sensation on our skin. It's essential to know that even if the air temperature is constant, the speed of the wind can make it feel much colder. The formula for wind chill incorporates this cooling effect.
The reason we experience a colder sensation is because the wind strips away the warm air that's enveloped close to our body. This makes the skin temperature lower, which our bodies interpret as being colder.

• Wind removes the insulating layer of warm air next to the skin.
• This results in increased heat loss and a cooler sensation.
• Thus, stronger winds make us feel colder, even if the temperature is the same.

So, the skin cooling effect, magnified by the wind, gives a perceived temperature that's lower than the actual air temperature.

Temperature Formulas

Temperature formulas like the wind chill formula help us quantify how cold it feels outside, taking into account factors like wind speed and air temperature. This provides us with a more accurate understanding of what conditions might feel like for our body.The wind chill formula we used is:\[ W(v, T) = 91.4 - \frac{(10.45 + 6.68 \sqrt{v} - 0.447 v)(457 - 5T)}{110} \]Here:

• \( v \) represents wind speed (miles per hour), showing how quick air moves past the skin.
• \( T \) is the thermometer-measured temperature (degrees Fahrenheit), the actual environment temperature.

Understanding these formulas allows us to better prepare for wind conditions by understanding how they affect our perceived temperature. Using the formula, we calculate a wind chill temperature that tells us what the air temperature would "feel like" due to wind's cooling effects.