Problem 88 - A Graphical Approach to Precalculus with Limits

Question

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 temperature is \( 62.6^{\circ}F \) at 3 miles altitude.

Step 1: Substitute the Value of x

First, insert the value of \( x \) into the function \( f(x) = 80 - 5.8x \). Here, we need to find \( f(3) \). So, substitute \( x = 3 \) into the function.

Step 2: Calculate Expression

Calculate \( f(3) = 80 - 5.8 \times 3 \). Multiply \( 5.8 \) by \( 3 \) to get \( 17.4 \).

Step 3: Perform Subtraction

Subtract \( 17.4 \) from \( 80 \). This gives \( 80 - 17.4 = 62.6 \).

Step 4: Interpret the Result

The value \( f(3) = 62.6 \) indicates that when the altitude is 3 miles, the air temperature is \( 62.6^{\circ}F \) under the given conditions.

Key Concepts

Understanding Functions

In mathematics, a function is like a machine that takes an input, performs a specific operation, and gives an output. Imagine you have a formula, like a recipe, that tells you how to mix ingredients to get a particular dish. Functions work in a similar way: they use a rule or a formula to transform given numbers (inputs) into new numbers (outputs). For instance, the function \( f(x) = 80 - 5.8x \) creates a connection between altitude (the input \(x\)) and temperature (the output \(f(x)\)).

Here are some key points about functions:

• The "\(x\)" is the independent variable, while "\(f(x)\)" is the dependent variable because its value depends on what \(x\) is.
• Each input \(x\) has one unique output \(f(x)\), ensuring that for every altitude, there’s a specific temperature.
• Functions help in modeling real-world situations, providing a simple and systematic way to predict changes.

Functions make it easier to understand complex systems by describing how variables like altitude and temperature interact.

Real-World Applications of Functions

Functions aren't just abstract mathematical concepts—they are powerful tools that solve real-world problems. In this exercise, the function \( f(x) = 80 - 5.8x \) helps us predict temperature changes as we go higher above the ground. This particular scenario relates to meteorology, which is the study of weather systems.

Real-world applications of functions include:

• Predicting temperature fluctuations at different altitudes for aviation and weather forecasts.
• Modeling financial growth where the input could be time, and the output could be profit or loss.
• Calculating distance over time in physics applications, such as speed or velocity functions.

Using functions simplifies decision-making processes and provides valuable insights in various fields from engineering to economics.

Delving Into Temperature Gradient

A temperature gradient describes the rate of temperature change over a certain distance or depth. In our exercise, the temperature gradient is \(5.8^{\circ} F\) per mile. This means for every mile you ascend, the temperature cools by \(5.8^{\circ} F\).

Important aspects of temperature gradients:

• They show how rapidly temperature changes over a distance, which helps predict environmental conditions.
• A larger gradient indicates a faster change, while a smaller gradient signals a slower change.
• Temperature gradients are crucial for understanding weather patterns and how they might affect various activities like hiking or flying.

By knowing the temperature gradient, we can better anticipate shifts in temperature as altitude changes, aiding planning and safety measures.

Exploring Altitude Effects on Temperature

The exercise highlights a common environmental phenomenon: as altitude increases, the temperature typically decreases. This is due to the thinning atmosphere at higher elevations, which holds less heat.

Here’s why altitude affects temperature:

• With higher altitude, air pressure drops, causing air to expand and its temperature to lower, a process called "adiabatic cooling."
• The thinner air at higher altitudes absorbs less heat from the Earth’s surface.
• Mountains and higher terrains, therefore, are usually cooler than low-lying areas, which is why we often see snow at high elevations even during warmer months.

Understanding the effects of altitude on temperature helps us prepare for temperature-related challenges we might face in various activities, from mountain climbing to aviation.