Problem 90 - A Graphical Approach to College Algebra

Question

90\. Air Temperature When the relative humidity is \(100 \%\) air cools \(5.8^{\circ} \mathrm{F}\) for every 1 -mile increase in altitude. If the temperature is \(80^{\circ} \mathrm{F}\) on the ground, then \(f(x)=80-5.8 x\) calculates the air temperature \(x\) miles above the ground. Find \(f(3)\) and interpret the result. (Source: Battan, L., Weather in Your Life, W.H. Freeman.)

Step-by-Step Solution

Answer

The air temperature 3 miles above the ground is 62.6°F.

Step 1: Understand the Function

The function given is \( f(x) = 80 - 5.8x \). This function calculates the air temperature at \( x \) miles above the ground when the relative humidity is 100%. The temperature drops \( 5.8^{\circ} \mathrm{F} \) for every 1-mile increase in altitude.

Step 2: Substitute the Value of x

The problem asks us to find \( f(3) \), which means we need to substitute \( x = 3 \) into the function \( f(x) = 80 - 5.8x \).

Step 3: Apply the Function

Substitute \( x = 3 \) into the function: \[f(3) = 80 - 5.8 \times 3\]Now, calculate the product: \[5.8 \times 3 = 17.4\]

Step 4: Calculate the Result

Subtract the product from \( 80 \): \[f(3) = 80 - 17.4 = 62.6\]

Step 5: Interpret the Result

The result \( f(3) = 62.6 \) indicates that the air temperature 3 miles above the ground is \( 62.6^{\circ}\mathrm{F} \) when the relative humidity is 100%.

Key Concepts

Altitude Effect on Temperature

Altitude plays a crucial role in determining the air temperature. As you ascend higher above ground level, temperatures typically decrease. This phenomenon occurs mainly because the atmosphere becomes less dense with increasing height. In our context, the challenge is to understand how altitude affects temperature when humidity is at 100%. According to the provided exercise, the temperature drops by \(5.8^{\circ} \mathrm{F}\) for every mile increase in altitude. This means as you climb, the air gets cooler, which is crucial for activities such as hiking or planning weather-related decisions.
Knowing how temperature changes with altitude helps in predicting weather conditions at higher elevations. While 5.8°F is a typical rate of temperature decrease, it's essential to note that extreme weather events can affect this rate. Having a grasp of this concept is vital for meteorological studies and practical applications like aviation and alpine navigation.

Linear Functions in Algebra

Linear functions in algebra provide a simple way to describe relationships between quantities. A linear function is expressed as \(f(x) = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In our exercise, the function \(f(x) = 80 - 5.8x\) describes how air temperature varies with altitude.
- **Slope \(m = -5.8\)**: This represents the rate of temperature decrease with altitude. A negative slope indicates a drop in temperature when going up.
- **Y-intercept \(b = 80\)**: This indicates the initial temperature at sea level or ground level when \(x = 0\).
Using linear functions allows us to compute values quickly and predict outcomes by substituting specific values. Here, it's crucial in climatology and environmental science, helping to understand and anticipate weather changes at different altitudes.

Temperature Calculation

Calculating the temperature at a different altitude is straightforward when using a linear function. For instance, in the given exercise, we're tasked to find \(f(3)\), which translates to finding the air temperature 3 miles above the ground.
By directly substituting \(x = 3\) into the function \(f(x) = 80 - 5.8x\), we get:

• Multiply the slope and the distance: \(5.8 \times 3 = 17.4\)


• Subtract this result from the initial temperature: \(80 - 17.4 = 62.6\)

The result, \(f(3) = 62.6^{\circ} \mathrm{F}\), provides the temperature at that altitude. This calculation is not only practical but also forms a foundation for understanding temperature dynamics in atmospheric studies. Applying such functions helps in aviation, climatology, and even preparing for outdoor activities that might involve significant elevation changes.