Problem 65 - Precalculus with Limits

Question

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. \(8 e^{-2 x / 3}=11\)

Step-by-Step Solution

Answer

The main solution of the given equation is the value of \(x\) which was initially estimated by the graphing tool and later confirmed by the algebraic verification. This specific value of \(x\) was obtained by using properties of exponential functions and applying appropriate logarithms.

Step 1: Use a Graphing Tool for Initial Approximation

First use an available graphing tool (for example, an online graphing calculator) and input the equation \(8 e^{-2 x / 3}-11=0\). It's adapted to have zero at one side for visualizing the intersection with the x-axis. On the graph, find the x-coordinate where the curve intersects the x-axis. Mark it down as an initial estimate.

Step 2: Approximate the Result to Three Decimal Places

From the graph, estimate the intersection point to the nearest three decimal places. This is the initial approximation of the solution.

Step 3: Verification Algebraically

Based on the properties of exponential functions, take the natural logarithm (ln) on both sides of the original equation: \(8 e^{-2 x / 3}=11\). This will modify the equation into a simpler form, namely \(-2 x / 3 = ln(11/8)\), which can be further simplified to \(x = -1.5*ln(11/8)\). Evaluating this expression should give a result that confirms the initial estimate from the graphing tool. If this is the case, then the solution is correct. If not, go back to step 1 and repeat the process.

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that can help us visualize and solve complex equations, like the exponential equation given in the exercise. These utilities, which include online graphing calculators and software, can graph functions and help you find intersections accurately.

When solving an equation like \(8 e^{-2x/3} = 11\) using a graphing utility, we rearrange it to \(8 e^{-2x/3} - 11 = 0\). This way, we can visualize it as a function intersecting the x-axis. The point where the graph crosses the x-axis corresponds to the solution of the equation.

Here are some steps to use a graphing utility effectively:

• Enter the equation into the graphing software.
• Look at the graph to find where it crosses the x-axis.
• Zoom in if needed to identify the x-intercept more precisely.

While the graph gives an approximate solution, it serves as a starting point for further algebraic verification. Overall, graphing utilities simplify the process of finding solutions to complex equations by providing a visual representation.

Natural Logarithm

A natural logarithm (\(\ln\)) is a logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. In solving exponential equations like \(8 e^{-2x/3} = 11\), the natural logarithm is used to transform the exponential part into a linear form, making it easier to solve algebraically.

Taking the natural logarithm of both sides of the equation changes it to \(-2x/3 = \ln(11/8)\). This transformation leverages the property that applying the logarithm to an exponential function allows the exponent to become a coefficient.

Here is why it is useful:

• The logarithm reverses the exponential function, simplifying solving.
• It helps extract x from the exponent in equations where it's otherwise locked inside.

By calculating \(x\) from \(-2x/3 = \ln(11/8)\), we can verify the solution obtained from the graphing tool. This step ensures the algebraic correctness and accuracy of our solution.

Exponential Functions

Exponential functions are mathematical expressions where the variable is in the exponent, such as \(e^{-2x/3}\). These functions describe processes that increase or decrease rapidly, like population growth or radioactive decay.

When dealing with an equation like \(8 e^{-2x/3} = 11\), understanding the behavior of exponential functions helps us anticipate how the graph will look and why specific solutions make sense.

Characteristics of exponential functions include:

• A constant base (like \(e\)), raised to the variable power.
• Rapid increase or decrease depending on the sign of the exponent.
• Graph transformations based on coefficient changes (e.g., \(8\) in \(8 e^{-2x/3}\)).

Exponential functions often require special methods like graphing and logarithmic transformation due to their unique nature. Grasping these concepts can demystify their behavior, making it easier to solve related equations.