Problem 162 - Calculus 3

Question

The apparent temperature index is a measure of how the temperature feels, and it is based on two variables: \(h,\) which is relative humidity, and \(t, \quad\) which is the air temperature. \(A=0.885 t-22.4 h+1.20 t h-0.544 .\) Find \(\frac{\partial A}{\partial t}\) and \(\frac{\partial A}{\partial h}\) when \(t=20^{\circ} \mathrm{F}\) and \(h=0.90\)

Step-by-Step Solution

Answer

\( \frac{\partial A}{\partial t} = 1.965 \), \( \frac{\partial A}{\partial h} = 1.6 \).

Step 1: Understand Partial Derivatives

In calculus, a partial derivative of a function of several variables is its derivative with respect to one of those variables, while the others are held constant. This exercise requires finding the partial derivatives of the apparent temperature index \( A \), with respect to temperature \( t \) and humidity \( h \).

Step 2: Find \( \frac{\partial A}{\partial t} \)

To calculate \( \frac{\partial A}{\partial t} \), differentiate the function \( A = 0.885t - 22.4h + 1.20th - 0.544 \) with respect to \( t \):1. Differentiate \( 0.885t \) with respect to \( t \) to get \( 0.885 \).2. \(-22.4h\) is treated as a constant, so its derivative is \( 0 \).3. Differentiate \( 1.20th \) with respect to \( t \) to get \( 1.20h \).4. The constant \(-0.544\) has a derivative of \( 0 \).Thus, \( \frac{\partial A}{\partial t} = 0.885 + 1.20h \).

Step 3: Evaluate \( \frac{\partial A}{\partial t} \) at the given point

Substitute \( h = 0.90 \) into the expression for \( \frac{\partial A}{\partial t} \): \[ \frac{\partial A}{\partial t} = 0.885 + 1.20(0.90) = 0.885 + 1.08 = 1.965 \]

Step 4: Find \( \frac{\partial A}{\partial h} \)

To calculate \( \frac{\partial A}{\partial h} \), differentiate the function \( A = 0.885t - 22.4h + 1.20th - 0.544 \) with respect to \( h \):1. \( 0.885t \) is treated as a constant, so its derivative is \( 0 \).2. Differentiate \(-22.4h\) with respect to \( h \) to get \(-22.4 \).3. Differentiate \( 1.20th \) with respect to \( h \) to get \( 1.20t \).4. The constant \(-0.544\) has a derivative of \( 0 \).Thus, \( \frac{\partial A}{\partial h} = -22.4 + 1.20t \).

Step 5: Evaluate \( \frac{\partial A}{\partial h} \) at the given point

Substitute \( t = 20 \) into the expression for \( \frac{\partial A}{\partial h} \): \[ \frac{\partial A}{\partial h} = -22.4 + 1.20(20) = -22.4 + 24 = 1.6 \]

Step 6: Conclude the Calculations

The partial derivatives at the point \( t = 20^{\circ} \mathrm{F} \) and \( h = 0.90 \) are \( \frac{\partial A}{\partial t} = 1.965 \) and \( \frac{\partial A}{\partial h} = 1.6 \).

Key Concepts

Apparent Temperature Index

Understanding the apparent temperature index is crucial for comprehending how temperature feels to us, beyond just the actual air temperature. It combines both air temperature and relative humidity to give a measure of perceived temperature. Much like how a wind chill factor makes cold air feel even colder, higher humidity levels with warm temperatures can make the air feel much hotter.

This concept is crucial because it helps us determine comfort levels in different environments and guide decisions for outdoor activities, dressing appropriately, and energy use in heating or cooling. The formula for apparent temperature gives us insight into how changes in either the air temperature or humidity can affect our comfort levels, making it an excellent example of applying partial derivatives.

Relative Humidity

Relative humidity is a measurement of the amount of water vapor present in the air compared to the maximum amount the air can hold at that temperature. It's expressed as a percentage, where a higher percentage indicates a higher level of moisture in the air. When relative humidity is high, it feels hotter than it actually is because the body struggles to evaporate sweat, its natural cooling mechanism.

In the context of the apparent temperature index, relative humidity directly influences how hot or cold the temperature feels. For instance, a day with 20°C temperature and 90% humidity will feel much warmer than a dry day at the same temperature. This concept is integral to understanding how partial derivatives can show us how changes in humidity affect the perceived temperature.

Air Temperature

Air temperature is the measure of how hot or cold the atmosphere is and is typically expressed in degrees Celsius or Fahrenheit. This measurement is fundamental as it directly influences weather predictions, comfort levels, energy usage for heating or cooling, and many other day-to-day activities.

In studies involving the apparent temperature index, air temperature acts as a critical variable. It combines with other factors like humidity to shape our experience of warmth or coolness. By using partial derivatives, we can determine how sensitive the apparent temperature is to changes in the actual air temperature, enhancing our understanding of climate conditions and helping us prepare better for unexpected weather changes. Understanding this interaction enables a deeper comprehension of both immediate comfort and broader environmental impacts.