Problem 62 - Algebra for College Students

Question

Find an approximate solution, to the nearest hundredth, for each of the following equations by graphing the appropriate function and finding the \(x\) intercept. (a) \(e^{x}=7\) (b) \(e^{x}=21\) (c) \(e^{x}=53\) (d) \(2 e^{x}=60\) (e) \(e^{x+1}=150\) (f) \(e^{x-2}=300\)

Step-by-Step Solution

Answer

(a) 1.95 (b) 3.04 (c) 3.97 (d) 4.09 (e) 2.71 (f) 7.90

Step 1: Understanding the Problem

We need to find approximate solutions to the given exponential equations by graphing their corresponding functions and identifying the x-intercepts of these graphs. The x-intercept occurs where the equation equals zero. Thus, we graph the functions derived from setting equations to zero.

Step 2: Set Equations to Zero

For each equation, rewrite them such that they equal zero. (a) \[ f(x) = e^x - 7 \](b) \[ f(x) = e^x - 21 \](c) \[ f(x) = e^x - 53 \](d)\[ f(x) = 2e^x - 60 \](e) \[ f(x) = e^{x+1} - 150 \](f) \[ f(x) = e^{x-2} - 300 \]

Step 3: Graph the Functions

Graph each function. In each graph: - The y-intercept is irrelevant; we are interested in the x-axis. - Focus on finding where the curve crosses the x-axis (this is our x-intercept).

Step 4: Find the X-intercepts

Identify the x-value where each graph crosses the x-axis, which gives:(a) For \( e^x - 7 \), the intercept is around 1.95.(b) For \( e^x - 21 \), the intercept is around 3.04.(c) For \( e^x - 53 \), the intercept is around 3.97.(d) For \( 2e^x - 60 \), the intercept is around 4.09.(e) For \( e^{x+1} - 150 \), rewrite it as \( e^{x} - 15 \), then the intercept is around 2.71.(f) For \( e^{x-2} - 300 \), rewrite it as \( e^{x} - 300e^2 \), then the intercept is around 7.90.

Step 5: Round to the Nearest Hundredth

Ensure each intercept is rounded to the nearest hundredth as required by the problem.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where the variable appears in the exponent or power. In problems such as

• \( e^x = 7 \)
• \( e^x = 21 \)

we look for the value of \( x \) that satisfies these equations when expressed as functions. Exponential equations can describe rapid growth and decay processes.

Understanding these equations involves recognizing the base (often \( e \), the natural base) and isolating the variable to solve for \( x \). Graphing these functions can visually demonstrate where solutions lie, particularly when they are set equal to constants, such as solving \( e^x = 7 \) by graphing \( f(x) = e^x - 7 \) and finding the intersection with the x-axis.

X-intercept

The x-intercept is a crucial concept when solving equations graphically. It is the point where a graph crosses the x-axis. At this intersection, the function value is zero. For example, in the function \( f(x) = e^x - 7 \), the x-intercept occurs where \( e^x = 7 \), which allows us to find the solution to the equation.

To identify the x-intercept of an exponential function:

• Graph the function in the coordinate plane.
• Locate where the graph meets the x-axis.
• The x-coordinate of this point is the solution to the equation.

By estimating the x-intercept, we can acquire approximate solutions without needing algebraic manipulation.

Function Graphing

Graphing functions is a pivotal technique in understanding how equations behave visually. It involves plotting points for the equation derived from setting functions equal to zero.

When graphing exponential functions, consider:

• Noticing the general shape: Exponential growth functions ascend rapidly, forming curves that increase steeply.
• The focus is on where the curve crosses the x-axis—this is our x-intercept.
• Graphs can be sketched manually or generated using graphing calculators or software.
  
  

The visual representation helps identify the approximate x-values where exponential equations reach their desired constant state, greatly aiding in intuitive problem solving.

Approximate Solutions

Sometimes, finding an exact solution algebraically isn't straightforward, particularly with exponential equations. That's where seeking approximate solutions using graphing is beneficial.

Approximate solutions offer a way to:

• Use the graph to find values close to exact solutions by viewing where the function approaches zero.
• Estimate to the nearest hundredth, providing practical solutions for real-world problems requiring precision.
• Employ tools like graphing software to zoom in on desired levels of accuracy.

These methods highlight how various approaches (analytical and graphical) can complement one another to provide understanding and practical problem-solving strategies.

Mathematical Problem Solving

Mathematical problem solving is an essential skill, involving the use of different strategies to find solutions effectively. For solving exponential equations using graphs, it integrates:

• Translating word problems or equations into graphical forms.
• Analyzing the graph to identify where the solutions lie, as shown through x-intercepts.
• Checking the accuracy of solutions and refining them if necessary by rounding.

This method of solving problems promotes deeper comprehension of mathematical concepts beyond mere number crunching, encouraging visual and intuitive learning. By integrating various techniques, students gain a robust toolkit for tackling diverse mathematical challenges.