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Chapter 3: Problem 42
Find the exact value of the logarithmic expression without using a calculator. (If this is not possible, state the reason.) $$2 \ln e^{6}-\ln e^{5}$$
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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 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.
The number \(N\) of trees of a given species per acre is approximated by the model \(N=68\left(10^{-0.04 x}\right), 5 \leq x \leq 40,\) where \(x\) is the average diameter of the trees (in inches) 3 feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when \(N=21\).
The numbers \(y\) of freestanding ambulatory care surgery centers in the United States from 2000 through 2007 can be modeled by \(y=2875+\frac{2635.11}{1+14.215 e^{-0.8038 t}}, \quad 0 \leq t \leq 7\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . (Source: Verispan) (a) Use a graphing utility to graph the model. (b) Use the trace feature of the graphing utility to estimate the year in which the number of surgery centers exceeded \(3600 .\)
The total interest \(u\) paid on a home mortgage of \(P\) dollars at interest rate \(r\) for \(t\) years is \(u=P\left[\frac{r t}{1-\left(\frac{1}{1+r / 12}\right)^{12 t}}-1\right]\) Consider a $$\$ 120,000$$ home mortgage at \(7 \frac{1}{2} \%\). (a) Use a graphing utility to graph the total interest function. (b) Approximate the length of the mortgage for which the total interest paid is the same as the size of the mortgage. Is it possible that some people are paying twice as much in interest charges as the size of the mortgage?
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