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Chapter 3: Problem 76

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

Short Answer

Expert verified
The solution to the equation can be found graphically and then confirmed algebraically. The exact numerical solution depends on the graph. After obtaining a numerical value for \(x\), plug it into the original equation to verify.

Step by step solution

01

Graph the equation

Use a graphing utility software to graph the given equation \(8 e^{-2 x / 3}=11\). The solution to the equation is where the graph intersects the x-axis.
02

Determine the approximate solution

By inspecting the graph and using the computer tool, determine the approximate x-value at which the graph intersects the x-axis. This will be the numerical solution to the equation.
03

Algebraic verification

Take the approximated value from step 2 and substitute it in the place of 'x' in the original equation. If both sides of the equation match approximately, the found solution is correct.

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Most popular questions from this chapter

An exponential growth model has the form ________ and an exponential decay model has the form ________.

A $$\$ 120,000$$ home mortgage for 30 years at \(7 \frac{1}{2} \%\) has a monthly payment of $$\$ 839.06 .$$ Part of the monthly payment is paid toward the interest charge on the unpaid balance, and the remainder of the payment is used to reduce the principal. The amount that is paid toward the interest is \(u=M-\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}\) and the amount that is paid toward the reduction of the principal is \(v=\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}\) In these formulas, \(P\) is the size of the mortgage, \(r\) is the interest rate, \(M\) is the monthly payment, and \(t\) is the time (in years). (a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 30 years of mortgage payments.) (b) In the early years of the mortgage, is the larger part of the monthly payment paid toward the interest or the principal? Approximate the time when the monthly payment is evenly divided between interest and principal reduction. (c) Repeat parts (a) and (b) for a repayment period of 20 years \((M=\$ 966.71) .\) What can you conclude?

Use the Richter scale \(R=\log \frac{l}{I_{0}}\) for measuring the magnitudes of earthquakes. Find the magnitude \(R\) of each earthquake of intensity \(I\) (let \(I_{0}=1\) ). (a) \(I=199,500,000\) (b) \(I=48,275,000\) (c) \(I=17,000\)

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1-\ln x}{x^{2}}=0$$

Find the exponential model \(y=a e^{b x}\) that fits the points shown in the graph or table. $$ \begin{array}{|l|l|l|} \hline x & 0 & 4 \\ \hline y & 5 & 1 \\ \hline \end{array} $$
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