Problem 14 - Algebra Form and Function

Question

Table 5.6 shows the air temperature \(T\) as a function of the height \(h\) above the earth's surface. \({ }^{6}\) Is \(T\) a linear function of \(h\) ? Give a formula if it is. $$ \begin{array}{l|r|r|r|r|r|r} \hline h, \text { height }(\mathrm{m}) & 0 & 2000 & 4000 & 6000 & 8000 & 10,000 \\ \hline T, \text { temperature }\left({ }^{\circ} \mathrm{C}\right) & 15 & 2 & -11 & -24 & -37 & -50 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Answer: Yes, the temperature (T) is a linear function of the height (h) above the Earth's surface. The formula representing this relationship is \(T(h) = \frac{-13}{2000} \cdot h + 15\).

Step 1: Analyze the given data

In the given table, we have values of height (h) and corresponding temperature (T). We will look at the differences in temperature and height as we progress along the table. $$ \begin{array}{l|r|r|r|r|r|r} \hline h, \text { height }(\mathrm{m}) & 0 & 2000 & 4000 & 6000 & 8000 &10,000 \\\ \hline T, \text { temperature }\left({ }^{\circ} \mathrm{C}\right) & 15 & 2 &-11 & -24 & -37 & -50 \\\ \hline \end{array} $$

Step 2: Compare the differences in T and h

Deduct consecutive temperature values and height values to see if the difference is constant or not. Temperature Difference: \(T2 - T1 = 2 - 15 = -13^\circ C\) \(T3 - T2 = -11 - 2 = -13^\circ C\) \(T4 - T3 = -24 - (-11) = -13^\circ C\) \(T5 - T4 = -37 - (-24) = -13^\circ C\) \(T6 - T5 = -50 - (-37) = -13^\circ C\) Height Difference: \(h2 - h1 = 2000 - 0 = 2000m\) \(h3 - h2 = 4000 - 2000 = 2000m\) \(h4 - h3 = 6000 - 4000 = 2000m\) \(h5 - h4 = 8000 - 6000 = 2000m\) \(h6 - h5 = 10000 - 8000 = 2000m\)

Step 3: Determine if T is a linear function of h and find the formula

Since the temperature difference is constant (-13) and the height difference is also constant (2000), we can conclude that T is a linear function of h. The slope of the linear function (m) can be calculated as the ratio of the temperature difference to the height difference: \(m = \frac{\Delta T}{\Delta h} = \frac{-13}{2000}= \frac{-13}{2000} m^{-1}\) To find the formula for the linear function, we can use the slope-intercept form of a linear function: \(T(h) = m \cdot h + b\) where b is the y-intercept (the temperature at h=0) From the table, we know the temperature is 15°C when height h is 0: \(15 = m \cdot 0 + b\) \(15 = b\) Now, we can plug the values of m and b into the equation to find the linear function: \(T(h) = \frac{-13}{2000} \cdot h + 15\)

Key Concepts

Slope

The slope of a linear function represents how one variable changes with respect to another. In a temperature function like the one we have, the slope tells us how much the temperature changes per unit of height.

Let's break this down:

• The slope is calculated by taking the difference in temperature between two points and dividing it by the difference in height between the same two points.
• In our example, the temperature decreases by 13°C for every 2000 meters increase in height.
• This gives us a slope of \(-\frac{13}{2000}\) m⁻¹.

A negative slope implies that as the height increases, the temperature decreases. This is typical in atmospheric data where temperatures often drop with elevation.

Temperature Function

A linear temperature function expresses temperature as a direct function of height. In this exercise, we have determined that temperature T is a linear function of height h.

The general form of a linear function is:
\[ T(h) = m \cdot h + b \] Here,

• \(m\) is the slope, which shows how temperature changes with height.
• \(b\) is the y-intercept, representing the temperature when the height is zero.

For this scenario, the intercept is 15°C, indicating the temperature at sea level. Together, these parameters form the linear equation \(T(h) = \frac{-13}{2000} \cdot h + 15\).

This formula helps predict temperatures at various heights by simply plugging in the height value.

Height and Temperature Relationship

Understanding the relationship between height and temperature involves recognizing how these two variables interact.

From the table:

• The consistent drop in temperature with increased height is a key characteristic of a linear relationship.
• Each increase of 2000 meters results in a temperature drop of 13°C.

These consistent changes imply a simple linear formula and demonstrate a predictable pattern in nature.

This relationship is important for understanding phenomena like weather changes and planning activities at different altitudes. Recognizing a linear relationship simplifies predictions, making it easier to anticipate the effects of changing height on temperature.