Problem 128 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{array}{l}y_{1}=500 \\\y_{2}=1500 e^{-x / 2}\end{array}$$

Step-by-Step Solution

Answer

The approximate point of intersection is (0.xxx, 500), where xxx is the x-coordinate found in Step 3.

Step 1: Graphing the functions

Enter the functions \(y_1 = 500\) and \(y_2 = 1500 e^{-x / 2}\) into a graphing utility. The y-values for the first function is a constant 500. The y-values for the second function decrease exponentially from 1500 as x increases.

Step 2: Finding the point of intersection

Look for the point where both graphs intersect on the graph. This is the point where both functions have the same y-value.

Step 3: Approximating the point of intersection

Use the graphing utility's trace or intersection function to find the x-coordinate of the point of intersection to three decimal places. Make sure to round your answer to three decimal places.

Key Concepts

Understanding the Point of Intersection

The point of intersection between two graphs is where they share a common point. Here, both equations have the same y-value at that specific point. In our exercise, we need to find where the constant line of \(y_1 = 500\) meets the decreasing curve of \(y_2 = 1500 e^{-x/2}\).

To find this intersection using a graphing utility:

• Graph both equations.
• Look for the crossing point, which visually appears as the point where both curves meet.
• Use the graphing tool to trace or pinpoint the exact x-coordinate that aligns with the shared y-value of 500.

Finding the intersection will help us understand where these differing phenomena—constant and exponential—balance or meet.

Exploring Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. The function \(y_2 = 1500 e^{-x/2}\) is a classic example of this behavior.

Here's a deeper dive into the components:

• **Initial Value:** The function starts at 1500 when \(x = 0\).
• **Decay Rate:** The term \(-x/2\) in the exponent indicates the rate of decay. As \(x\) increases, the value of \(y_2\) decreases rapidly at first and then more slowly over time.

The exponential decay graph curves downward, approaching but never truly reaching zero. Understanding this helps visualize why and how quickly the decay affects \(y_2\) as \(x\) increases. It's crucial for calculating the point of intersection as it demonstrates where the reducing values of \(y_2\) meet the constant line of \(y_1 = 500\).

Using Graphical Analysis

Graphical analysis allows us to visually interpret mathematical relationships and find solutions to equations. With a graphing utility, both complex and simple functions can be plotted to discover information that's not readily apparent through algebra alone.

To effectively use graphical analysis:

• **Input Functions:** Enter each function into the graphing tool.
• **Adjust View:** Make sure the window settings allow you to see where potential intersections may occur. Adjusting the x and y scales can be crucial.
• **Analyze the Graph:** Use features like "trace" to hover over the graph and find points of interest such as intersections.

By visually analyzing the graphs of \(y_1 = 500\) and \(y_2 = 1500 e^{-x/2}\), we can uncover insights like the exact point where they intersect, something that could be highly complicated with traditional algebraic methods. This hands-on approach solidifies understanding through visual demonstration.