Problem 40 - Prealgebra

Question

On a certain day, the temperature on the ground is \(72^{\circ}\) Fahrenheit, and the temperature \(T\) at an altitude of \(A\) feet above the ground is given by the equation \(T=72-\frac{1}{300} A\). Graph this equation on the rectangular coordinate system here. Note that we have restricted our graph to positive values of \(A\) only. (GRAPH CANT COPY)

Step-by-Step Solution

Answer

Graph a line from (0, 72) to (21600, 0) on a coordinate plane.

Step 1: Understand the Equation

The equation given is \(T = 72 - \frac{1}{300} A\). This represents the temperature \(T\) at an altitude \(A\) feet above the ground. It is a linear equation in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Identify the Key Components

In the equation \(T = 72 - \frac{1}{300} A\), the y-intercept is 72 because when \(A = 0\), \(T = 72\). The slope \(-\frac{1}{300}\) indicates that the temperature decreases by \(\frac{1}{300}\) degree Fahrenheit for every foot increase in altitude.

Step 3: Determine the X and Y Intercepts

The y-intercept occurs when \(A = 0\), giving \(T = 72\). To find the x-intercept (where the temperature would be 0), set \(T = 0\) and solve for \(A\): \(0 = 72 - \frac{1}{300}A\). Solving this, we get \(A = 21600\) feet.

Step 4: Plot Key Points

Plot the y-intercept (0, 72) on the graph. Then plot the x-intercept (21600, 0) found earlier. These points will be enough to draw the line accurately since the line is straight between them.

Step 5: Draw the Line

Draw a straight line through the plotted points (0, 72) and (21600, 0). This line represents the equation \(T = 72 - \frac{1}{300}A\) for \(A > 0\).

Key Concepts

Graphing Linear Functions

Graphing linear functions is an essential skill in understanding relationships between variables. A linear function is typically expressed in the form of an equation such as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept. The slope determines how steep the line is and indicates how much \( y \) changes for every unit increase in \( x \). The y-intercept is the point where the line crosses the y-axis.

To graph a linear function, follow these basic steps:

• Identify the slope and y-intercept from the equation.
• Locate the y-intercept on the graph. This is your starting point.
• Use the slope to find another point on the line. For instance, if your slope is \( -\frac{1}{300} \), go down 1 unit in \( y \) for every 300 units you move to the right in \( x \).
• Draw a straight line through these points, extending it in both directions.

By positioning the line on the graph thoughtfully, you can visualize the relationship between two variables and understand how a change in one affects the other.

Temperature vs Altitude

The relationship between temperature and altitude is an interesting example of a linear relationship. In the provided exercise, this is illustrated with the equation \( T = 72 - \frac{1}{300} A \), where \( T \) is the temperature in degrees Fahrenheit, and \( A \) is the altitude in feet.
As altitude increases, temperature decreases consistently at a rate determined by the slope \( -\frac{1}{300} \). This means for every increase of 300 feet in altitude, the temperature decreases by 1 degree Fahrenheit. This negative slope shows a downward trend, which can be graphed accordingly.

In real-life applications, understanding this relationship can help in predicting weather conditions or preparing for changes in temperature when traveling to different altitudes. It gives us a straightforward way to understand how environmental factors interact through a simple linear model.

Coordinate Systems

To effectively graph functions and relationships like the temperature-altitude equation, one must have a basic understanding of coordinate systems. The rectangular coordinate system, or Cartesian plane, is a two-dimensional plane used for graphing these types of equations.
This system consists of:

• An x-axis, which runs horizontally and usually represents the independent variable such as altitude \( A \).
• A y-axis, which runs vertically and typically represents the dependent variable such as temperature \( T \).

The point where these axes intersect is called the origin, located at (0,0). Each point on the graph is defined by a pair of numbers (x, y), indicating its position relative to the x and y axes. In plotting graphs, it's crucial to accurately place points such as the y-intercept and x-intercept, which provide critical reference points for the line.

Learning how to navigate and plot points in a coordinate system is vital for accurately representing and understanding mathematical relationships in graph form.