Problem 86 - Algebra and Trigonometry

Question

Air Temperature As dry air moves upward, it expands and, in so doing, cools at a rate of about \(1^{\circ} \mathrm{C}\) for each 100 -meter rise, up to about 12 \(\mathrm{km}\) . (a) If the ground temperature is \(20^{\circ} \mathrm{C},\) write a formula for the temperature at height \(h\) . (b) What range of temperatures can be expected if an airplane takes off and reaches a maximum height of 5 \(\mathrm{km} ?\)

Step-by-Step Solution

Answer

Formula: \( T(h) = 20 - 10h \). Temperature range: 20°C to -30°C for heights 0-5 km.

Step 1: Understanding the Problem

We know that the temperature decreases as the altitude increases. Given the ground temperature and a rate at which the temperature decreases per 100 meters, we need to express this as a formula and find the temperature range up to 5 km.

Step 2: Finding the Formula for Temperature at Height h

Let the ground temperature be 20°C. The temperature decreases by 1°C for each 100 meters, which is the same as 0.01 km. Therefore, for a height \( h \) in km, the decrease in temperature is \( h \times 10 \). The formula for the temperature \( T(h) \) is given by:\[ T(h) = 20 - 10h \]

Step 3: Calculating Temperature at Maximum Height

The maximum height mentioned is 5 km. Substitute \( h = 5 \) into the formula to find the temperature at this height:\[ T(5) = 20 - 10 \times 5 = 20 - 50 = -30 \text{°C} \]

Step 4: Finding the Range of Temperatures

At the ground level (\( h = 0 \)), the temperature is 20°C. At the maximum height of 5 km, the temperature is -30°C. Thus, the range of temperatures from the ground to 5 km is from 20°C to -30°C.

Key Concepts

Temperature Gradient

A temperature gradient refers to the rate of change in temperature relative to distance in a specific direction. When discussing the atmosphere, this gradient often relates to altitude. As air rises in the atmosphere, it expands and cools. In the context of the provided exercise, the temperature gradient is a decrease of 1°C for every 100 meters or 0.1 kilometers the air rises. This means that if you start at 20°C at ground level, the temperature drops by 1°C per 100 meters, forming a predictable pattern or formula for temperature at different heights. Understanding this gradient is essential for calculating temperature changes during vertical travel, like an airplane's ascent.

Linear Equations

Linear equations are mathematical expressions that describe a straight-line relationship between two variables. In the context of our temperature exercise, we're using a linear equation to express how temperature changes with altitude. The general form of a linear equation is:

• y = mx + b

Here, 'y' is the dependent variable (temperature), 'm' is the slope (temperature gradient, -10 in this case), 'x' is the independent variable (height in km), and 'b' is the y-intercept (ground temperature, 20°C). The specific equation is:

• \( T(h) = 20 - 10h \)

This equation allows us to calculate the temperature at any given height 'h', showing the power of linear equations in environmental calculations.

Environmental Science

Environmental science often deals with understanding how different factors such as temperature, air pressure, and atmospheric conditions interact. This exercise, though primarily a mathematical problem, ties directly into larger environmental concepts. In the real world, knowing the temperature at various altitudes is crucial for understanding weather patterns, climate processes, and aviation safety. The concept of temperature gradient plays a big role in phenomena such as cloud formation, the behavior of wind currents, and even the distribution of life in ecosystems. By studying these gradients and equations, we gain insight into the fundamental processes that govern our planet's atmosphere and help predict changes that can affect everything from local weather to global climate.