Problem 51 - Calculus for Biology and Medicine

Question

In Section \(10.1\), Problem 3 , we introduced wind chill as a way of calculating the apparent temperature a person would feel as a function of the real air temperature, \(T\), and \(V\) in mph. Then the wind chill (i.e., the apparent temperature) is: $$\begin{array}{c}W(T, V)=35.74+0.6215 T-35.75 V^{0.16} \\\0.4275 T V^{0.16}\end{array}$$ (a) By calculating the appropriate partial derivative, show that increasing \(T\) always increases \(W\). (b) Under what conditions does increasing \(V\) decrease \(W\) ? Your answer will take the form of an inequality involving \(T\). (c) Assuming that \(W\) should always decrease when \(V\) is increased, use your answer from (b) to determine the largest domain in which this formula for \(W\) can be used.

Step-by-Step Solution

Answer

(a) Increasing \( T \) always increases \( W \). (b) Increasing \( V \) decreases \( W \) when \( T > 83.63 \). (c) The largest domain is \( T > 83.63 \).

Step 1: Calculate Partial Derivative with Respect to T

To find how the wind chill changes with respect to temperature, calculate the partial derivative of \( W(T, V) \) with respect to \( T \):\[ \frac{\partial W(T, V)}{\partial T} = 0.6215 + 0.4275 V^{0.16} \]. Since both 0.6215 and 0.4275 are positive constants and \( V^{0.16} \) is always positive for any wind speed, the derivative is always positive. This shows increasing \( T \) always increases \( W \).

Step 2: Calculate Partial Derivative with Respect to V

Now, calculate the partial derivative with respect to \( V \) to understand how wind speed affects the wind chill: \[ \frac{\partial W(T, V)}{\partial V} = -35.75 \, \cdot \, 0.16 \, \cdot \, V^{-0.84} + 0.4275 \, T \, \cdot \, 0.16 \, \cdot \, V^{-0.84} \]. Simplify it to: \[ \frac{\partial W(T, V)}{\partial V} = V^{-0.84} (-5.72 + 0.0684 T) \].

Step 3: Find Conditions for V to Decrease W

For \( \frac{\partial W(T, V)}{\partial V} < 0 \), the expression \( V^{-0.84} (-5.72 + 0.0684 T) \) must be negative. Since \( V^{-0.84} \) is always positive, it requires \( -5.72 + 0.0684 T < 0 \), which simplifies to \( T > \frac{5.72}{0.0684} \approx 83.63 \). Thus, \( T > 83.63 \) ensures increasing \( V \) reduces \( W \).

Step 4: Determine Largest Domain for Decreasing W with Increasing V

From Step 3, to ensure wind chill decreases with increasing wind speed, \( T \) must be over 83.63 degrees. Therefore, the formula for \( W \) can be applied appropriately only when \( T > 83.63 \).

Key Concepts

Partial Derivatives

In calculus, a partial derivative is a derivative taken of a function with respect to one variable, while holding other variables constant. This is particularly helpful in multivariable functions where different variables affect the outcome. For instance, the wind chill formula depends on two variables: air temperature, \(T\), and wind speed, \(V\).

Calculating a partial derivative allows us to see how changes in one factor influence the entire function. When we compute the partial derivative of the wind chill, \( W(T, V) \), with respect to the temperature \( T \), we get:

• \( \frac{\partial W(T, V)}{\partial T} = 0.6215 + 0.4275 V^{0.16} \)

Both terms in this equation are positive, making the whole expression positive. This tells us that as temperature \( T \) rises, \( W \), or the apparent temperature, also increases.

Similarly, the partial derivative of \( W \) with respect to wind speed \( V \) helps us analyze how \( V \) affects the wind chill, enabling us to derive conditions for when a change in wind speed results in a decrease in apparent temperature.

Wind Chill

Wind chill is a concept in meteorology that describes how the wind and air temperature combine to create the apparent temperature a person actually feels. This can differ significantly from the actual air temperature due to the cooling effect of the wind on our skin.

The formula for wind chill we are examining is:

• \(W(T, V) = 35.74 + 0.6215 T - 35.75 V^{0.16} + 0.4275 T V^{0.16}\)

This equation incorporates both temperature \( T \) and wind speed \( V \), allowing us to determine how they interact to produce the chilling effect. When wind speed increases, it can draw heat away from the body more quickly than still air, which makes it feel colder than it actually is.

Understanding this formula helps us predict how variations in temperature and wind speed will affect the perceived coldness, aiding in preparing individuals for cold weather conditions.

Inequalities

In the context of this problem, inequalities help define the conditions under which the wind chill decreases as the wind speed increases. This involves using inequalities to frame conditions that ensure the derivative of the wind chill relative to wind speed is negative.

From our calculations, when solving:

• \( -5.72 + 0.0684 T < 0 \)

We isolate \( T \) to obtain \( T > 83.63 \). This inequality indicates a vital threshold: when the temperature \( T \) is above 83.63 degrees, increasing the wind speed results in a reduced wind chill.

Using inequalities like these provides clear parameters and helps establish the range within which the wind chill formula is reliable and applicable, particularly under specific weather conditions. Understanding these mathematical conditions is crucial for practical real-world applications, such as determining appropriate clothing for cold weather.