Problem 58 - Precalculus : Building Concepts and Connections

Question

Use a graphing utility to solve each equation for \(x.\) $$25=50 e^{-0.05 x}$$

Step-by-Step Solution

Answer

The solution to the equation \(25 = 50 e^{-0.05x}\) will be found at the x-intercept of the graph of the equation \(50e^{-0.05x} - 25 = 0\). The exact solution may vary slightly depending on the precision of the graphing tool used.

Step 1: Set up the equation to graph

Format the equation to work with a graphing utility. You can do this by moving everything to one side to produce: \(50e^{-0.05x} - 25 = 0\) . This way, where this equation equals zero, we can find the x.

Step 2: Graph the equation

Use a graphing utility to graph the equation \(50e^{-0.05x} - 25\). Most calculators or online tools offer graphing capabilities.

Step 3: Find the x-intercept

The x-intercept is the value of \(x\) when the equation equals zero. To do this, find the point at which the graph of your equation crosses the x-axis. This will be the solution to your equation.

Key Concepts

Understanding Exponential Equations

Exponential equations involve variables in the exponent and are a key part of mathematics, especially in modeling growth and decay. In our specific example, the equation is set up as \(25 = 50e^{-0.05x}\). Here, \(e\) represents Euler's number, a constant approximately equal to 2.71828. This constant is widely used in real-world calculations, such as compound interest and population growth.

To solve exponential equations, you often need to be familiar with manipulating expressions and working with logarithms. These equations can usually be rewritten by isolating the exponential part, as shown in our exercise where the equation is reformatted to \(50e^{-0.05x} - 25 = 0\). Once you have this setup, you can use various tools such as graphing utilities to visually or numerically find solutions.

Finding the X-Intercept

The x-intercept is crucial when solving equations involving graphs. It's essentially the point where the graph of an equation crosses the x-axis. For exponential equations like the one in our exercise \(50e^{-0.05x} - 25 = 0\), the x-intercept provides the solution for the variable \(x\).

To find the x-intercept, you will first need to graph the equation. This is typically done using graphing tools or calculators. The intersection with the x-axis represents the value of \(x\) for which the equation equals zero. In practical terms, it's the input value where the output is zero. By finding this point, you solve the exercise's core challenge—determining the x value that satisfies the original equation.

Using Graphing Calculators

Graphing calculators are powerful tools that assist in solving various types of mathematical equations, including exponential equations. They can graph complex functions and make finding solutions such as the x-intercept relatively straightforward.

To use a graphing calculator effectively, you should first input the equation in the correct format, like \(50e^{-0.05x} - 25\). Most graphing calculators provide functions for entering exponents and constants like \(e\).

Once graphed, you can visually inspect where the graph crosses the x-axis, which corresponds to the x-intercept. Many graphing calculators have features to help accurately determine these points, sometimes labeled as "calculate zeros" or "find roots." This technology makes solving even tough equations manageable and allows you to understand graph behavior better. Here’s how using graphing calculators simplifies the process:

• Visual representation of the equation
• Accurate identification of intercepts
• Quick recalculations for adjustments

Working with these tools not only aids in solving the problem at hand but also develops a deeper understanding of exponential functions and their graphical trends.