Problem 66
Question
Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. \(e^{0.09 t}=3\)
Step-by-Step Solution
Verified Answer
By graphing the function, a value for \(t\) that approximates to 12.206 is found. When calculated algebraically, the exact value for \(t\) is found as \(t = ln(3)/0.09\), which approximates to 12.206, therefore, the graphical solution and the algebraic solution are consistent.
1Step 1: Graphical Approach
Use a graphing utility to plot the graph of the function \(y= e^{0.09 t}-3\). The points where the curve intersects the x-axis are the approximate solutions to the original equation. This is because when the function equals zero, it means that \(e^{0.09 t}=3\).
2Step 2: Approximate Solution
Observe where the graph intersects the x-axis to approximate the solution. Round the result to three decimal places.
3Step 3: Algebraic Verification
To verify the result, substitute the approximated \(t\) value into the original equation \(e^{0.09 t}=3\). If the left hand side of the equation evaluates approximately to 3, then the approximation is likely correct.
4Step 4: Solving algebraically
To find the exact solution, take the natural logarithm (ln) of both sides of the equation.This can make the exponent on the left hand side to come down, transforming the equation into \(0.09 t = ln(3)\). Then, isolate \(t\), \(t = ln(3)/0.09\)
Key Concepts
Understanding Graphing UtilitiesThe Power of Natural LogarithmsExponential Functions: The EssentialsAlgebraic Verification as a Sealing Step
Understanding Graphing Utilities
Graphing utilities, such as graphing calculators or software programs, are essential tools for visualizing and solving mathematical problems, especially those involving exponential equations. By entering an equation into a graphing utility, students can quickly see its shape and identify important features like intercepts and asymptotes.
When dealing with the equation like \(e^{0.09 t}=3\), the graphing utility allows pupils to plot the function \(y= e^{0.09 t}-3\). Why subtract 3? This is because finding where this new function crosses the x-axis (\(y=0\)) gives us the solutions for the original equation. It's a visual representation that bridges conceptual understanding with the actual solutions to the problems.
When dealing with the equation like \(e^{0.09 t}=3\), the graphing utility allows pupils to plot the function \(y= e^{0.09 t}-3\). Why subtract 3? This is because finding where this new function crosses the x-axis (\(y=0\)) gives us the solutions for the original equation. It's a visual representation that bridges conceptual understanding with the actual solutions to the problems.
Advantages of Using a Graphing Utility
- Instant visualization of complicated functions.
- Helpful in predicting the number of solutions and their approximate values.
- Facilitates understanding of the function's behavior over different intervals.
The Power of Natural Logarithms
The natural logarithm, denoted as \(ln\), is a mathematical function that is the inverse of the exponential function \(e^x\). In simpler terms, if \(e^x = y\), then \(ln(y) = x\). It plays a vital role in unraveling exponential equations, which otherwise seem daunting due to their non-linear nature.
In the context of solving \(e^{0.09 t}=3\), using the natural logarithm allows for the exponents to be brought down to the base level, enabling further manipulation and simplification. It transforms the equation into a linear form \(0.09t = ln(3)\), which can be easily tackled using basic algebra.
In the context of solving \(e^{0.09 t}=3\), using the natural logarithm allows for the exponents to be brought down to the base level, enabling further manipulation and simplification. It transforms the equation into a linear form \(0.09t = ln(3)\), which can be easily tackled using basic algebra.
Why Natural Logarithms Are Unique
- They are the default choice for solving equations with base \(e\).
- They provide a way to linearize exponential relationships.
- Understanding \(ln\) is crucial for higher-level calculus and beyond.
Exponential Functions: The Essentials
Exponential functions represent continuous growth or decay processes and are commonly written in the form \(y=a^x\) where \(a\) is the base and \(x\) is the exponent. When the base is the irrational number \(e\), which is approximately 2.71828, the function describes natural growth or decay and has unique properties exploited in calculus and real-world applications.
The function \(e^{0.09t}\) is an exponential function, and solving it requires strategies that can handle the variable exponent. Utilizing the natural logarithm caters specifically to this challenge. Graphing this function would typically show a curve rising rapidly, indicating the accelerating nature of exponential growth.
The function \(e^{0.09t}\) is an exponential function, and solving it requires strategies that can handle the variable exponent. Utilizing the natural logarithm caters specifically to this challenge. Graphing this function would typically show a curve rising rapidly, indicating the accelerating nature of exponential growth.
Key Characteristics of Exponential Functions
- Rapid increase or decrease (growth or decay)
- Have a constant rate of change on a logarithmic scale
- Frequently occur in real-life scenarios like population growth, radioactive decay, and financial interest calculations
Algebraic Verification as a Sealing Step
Algebraic verification is a crucial final check to ensure the solutions obtained from a graphing utility or other methods are indeed correct. This process involves substituting the found or approximated values back into the original equation to see if the equation holds true.
In the given problem, after approximating \(t\) from the graph, it's substituted back into \(e^{0.09 t}=3\). If the left side closely equals 3, it reaffirms confidence in the approximation. Algebraic verification serves as a seal of accuracy, preventing the overlook of errors or misinterpretations made during the problem-solving process.
In the given problem, after approximating \(t\) from the graph, it's substituted back into \(e^{0.09 t}=3\). If the left side closely equals 3, it reaffirms confidence in the approximation. Algebraic verification serves as a seal of accuracy, preventing the overlook of errors or misinterpretations made during the problem-solving process.
Significance of Algebraic Verification
- It corroborates the solutions from graphical or numerical methods.
- Offers a mathematical proof that the solutions are valid.
- Enhances understanding and ensures mastery of the concepts involved.
Other exercises in this chapter
Problem 65
Use your school's library, the Internet, or some other reference source to write a paper describing John Napier's work with logarithms.
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Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. \(h(x)=\ln (x+5)\)
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