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Chapter 3: Problem 22
Solve for \(x\). $$\log x=-2$$
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The amount of time (in hours per week) a student utilizes a math-tutoring center roughly follows the normal distribution \(y=0.7979 e^{-(x-5.4)^{2} / 0.5},\) \(4 \leq x \leq 7,\) where \(x\) is the number of hours. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average number of hours per week a student uses the tutoring center.
The populations \(P\) (in thousands) of Horry County, South Carolina from 1970 through 2007 can be modeled by \(P=-18.5+92.2 e^{0.0282 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 1970\. (Source: U.S. Census Bureau) (a) Use the model to complete the table. $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2007 \\ \hline \text { Population } & & & & & \\ \hline \end{array} $$ (b) According to the model, when will the population of Horry County reach \(300,000 ?\) (c) Do you think the model is valid for long-term predictions of the population? Explain.
If $$\$ 1$$ is invested in an account over a 10-year period, the amount in the account, where \(t\) represents the time in years, is given by \(A=1+0.075 \llbracket t \rrbracket\) or \(A=e^{0.07 t}\) depending on whether the account pays simple interest at \(7 \frac{1}{2} \%\) or continuous compound interest at \(7 \%\). Graph each function on the same set of axes. Which grows at a higher rate? (Remember that \(\llbracket t \rrbracket\) is the greatest integer function discussed in Section 1.6.)
Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1+\ln x}{2}=0$$
A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers \(y\) of cell sites from 1985 through 2008 can be modeled by \(y=\frac{237,101}{1+1950 e^{-0.355 t}}\) where \(t\) represents the year, with \(t=5\) corresponding to \(1985 .\) (Source: CTIA-The Wireless Association) (a) Use the model to find the numbers of cell sites in the years 1985,2000 , and 2006 . (b) Use a graphing utility to graph the function. (c) Use the graph to determine the year in which the number of cell sites will reach 235,000 . (d) Confirm your answer to part (c) algebraically.
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