Problem 3 - Algebra Form and Function

Question

The temperature of the soil is \(30^{\circ} \mathrm{C}\) at the surface and decreases by \(0.04^{\circ} \mathrm{C}\) for each centimeter below the surface. Express temperature \(T\) as a function of depth \(d\), in centimeters, below the surface.

Step-by-Step Solution

Answer

Answer: T(d) = -0.04d + 30

Step 1: Identify the initial temperature

The initial temperature (\(T_0\)) is given at the surface, which is \(30^{\circ} \mathrm{C}\).

Step 2: Determine the slope

The rate at which the temperature decreases for every centimeter below the surface is constant, and it is given as \(0.04^{\circ} \mathrm{C}\) per centimeter. Since temperature decreases as we go deeper, the slope will be negative, so the slope (m) is \(-0.04^{\circ} \mathrm{C}\) per centimeter.

Step 3: Define the linear function

We can express the temperature (T) as a linear function of depth (d) using the general form of a linear equation: \(T = m \cdot d + T_0\). Here, m is the slope and \(T_0\) is the initial temperature.

Step 4: Substitute the slope and initial temperature

By substituting the slope and initial temperature into the linear equation, we get the function that describes the temperature as a function of depth: \(T(d) = -0.04d + 30\). Now, we have the desired function that models the temperature (T) based on the depth (d) below the surface.

Key Concepts

Temperature Gradient

The term "temperature gradient" describes how temperature changes over a specific distance. In our soil example, the temperature gradient speaks to the change in soil temperature as we move deeper beneath the surface. Here, the gradient is expressed as \(-0.04^{\circ} \text{C per centimeter}\). This means for every centimeter you delve into the soil, the temperature decreases by \(-0.04^{\circ} \text{C}\). Remember, a negative sign in this context indicates a decrease in temperature. For students working with temperature gradients, it's useful to:

• Identify whether the gradient is increasing or decreasing.
• Consider the physical implications of these changes in temperature, be it in soil, water, or air.
• Use units like °C/cm to clearly understand the rate of change.

These gradients are crucial in determining how quickly the environment changes, impacting everything from weather prediction to understanding soil behavior.

Depth and Temperature Relationship

The relationship between depth and temperature is linear and directly represented by our equation \(T(d) = -0.04d + 30\). As depth \(d\) increases, the temperature \(T\) systematically decreases. This relationship highlights one of the crucial aspects of linear equations: each unit change in \(d\) results in a consistent and predictable change in temperature.Visualizing this:

• The starting point or y-intercept is at the surface temperature of \(30^{\circ} \text{C}\).
• With increasing depth, the temperature drops at a steady rate indicated by the slope \(-0.04^{\circ} \text{C per centimeter}\).
• This linear pattern holds true unless other environmental conditions cause deviations.

Such dependencies between depth and temperature have applications in geology, oceanography, and environmental science.

Linear Equations in Real-World Contexts

Linear equations often find themselves at the heart of real-world problem-solving because of their simplicity and predictability. In many situations, like our soil temperature example, linear equations model natural phenomena where there is a constant rate of change. Real-world scenarios where linear equations are applicable include:

• Weather prediction models that guess temperature changes over time or space.
• Economics, where linear relationships model supply and demand or cost predictions based on time.
• Physics, describing constant velocity motion where distance increases linearly with time.

Understanding these relationships is essential because it allows us to predict outcomes accurately. Mastery of forming and interpreting these equations helps in analyzing trends anywhere a steady rate is present, simplifying complex data into manageable, understandable forms.