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Chapter 3: Problem 66

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$e^{0.09 t}=3$$

Short Answer

Expert verified
The approximate solution to the equation \(e^{0.09t}=3\) is \(t = 11.513\). The validity of this solution has been confirmed algebraically.

Step by step solution

01

Graph the Equation

Sketch the equation \(e^{0.09t}=3\). This will allow visualization of the problem to give an approximate value. Input the function \(e^{0.09t}\) to one graphing calculator and the constant function \(3\) to another. Look for the point of intersection which gives the solution to the problem.
02

Approximate Solution

From the graph, observe when the value of the function \(e^{0.09t}\) is equal to \(3\). This happens at approximately \(t = 11.513\). So an approximate solution to the problem based on the graph is \(t = 11.513\).
03

Algebraic Verification

To verify the solution algebraically, replace \(t\) with the approximate solution in the original equation. So, \(e^{0.09 * 11.513}\) should approximately yield 3. Doing the calculation gives us a value of \(2.999\), which is equal to \(3\) rounded to three decimal places, confirming the validity of the solution.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log (3 x+4)=\log (x-10)$$

Data Analysis The table shows the time \(t\) (in seconds) required for a car to attain a speed of \(s\) miles per hour from a standing start. $$\begin{array}{|c|c|} \hline \text { Speed, S } & \text { Time, t } \\ \hline 30 & 3.4 \\ 40 & 5.0 \\ 50 & 7.0 \\ 60 & 9.3 \\ 70 & 12.0 \\ 80 & 15.8 \\ 90 & 20.0 \\ \hline \end{array}$$ Two models for these data are as follows. \(t_{1}=40.757+0.556 s-15.817 \ln s\) \(t_{2}=1.2259+0.0023 s^{2}\) (a) Use the regression feature of a graphing utility to find a linear model \(t_{3}\) and an exponential model \(t_{4}\) for the data. (b) Use the graphing utility to graph the data and each model in the same viewing window. (c) Create a table comparing the data with estimates obtained from each model. (d) Use the results of part (c) to find the sum of the absolute values of the differences between the data and the estimated values given by each model. Based on the four sums, which model do you think best fits the data? Explain.

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In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 4} x$$
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