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Chapter 3: Problem 36
Solve the exponential equation algebraically. Approximate the result to three decimal places. $$4 e^{x}=91$$
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The populations \(P\) (in thousands) of Pittsburgh, Pennsylvania from 2000 through 2007 can be modeled by \(P=\frac{2632}{1+0.083 e^{0.0500 t}}\) where \(t\) represents the year, with \(t=0\) corresponding to \(2000 .\) (Source: U.S. Census Bureau) (a) Use the model to find the populations of Pittsburgh in the years \(2000,2005,\) and 2007 . (b) Use a graphing utility to graph the function. (c) Use the graph to determine the year in which the population will reach 2.2 million. (d) Confirm your answer to part (c) algebraically.
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 logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x+\ln (x+1)=1$$
Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$\ln (x+1)=2-\ln x$$
Complete the table for the time \(t\) (in years) necessary for \(P\) dollars to triple if interest is compounded annually at rate \(r\). $$ \begin{array}{|l|l|l|l|l|l|l|} \hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% \\ \hline t & & & & & & \\ \hline \end{array} $$
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