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Chapter 3: Problem 81
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x=-3$$
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Find the exponential model \(y=a e^{b x}\) that fits the points shown in the graph or table. $$ \begin{array}{|l|l|l|} \hline x & 0 & 4 \\ \hline y & 5 & 1 \\ \hline \end{array} $$
In a group project in learning theory, a mathematical model for the proportion \(P\) of correct responses after \(n\) trials was found to be \(P=0.83 /\left(1+e^{-0.2 n}\right)\) (a) Use a graphing utility to graph the function. (b) Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of this problem. (c) After how many trials will \(60 \%\) of the responses be correct?
After discontinuing all advertising for a tool kit in \(2004,\) the manufacturer noted that sales began to drop according to the model \(S=\frac{500,000}{1+0.4 e^{k t}}\) where \(S\) represents the number of units sold and \(t=4\) represents \(2004 .\) In \(2008,\) the company sold 300,000 units. (a) Complete the model by solving for \(k\). (b) Estimate sales in 2012 .
Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$2 \ln (x+3)=3$$
Complete the table for the time \(t\) (in years) necessary for \(P\) dollars to triple if interest is compounded continuously 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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