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Chapter 3: Problem 15
Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{3} 7$$
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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.
The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 250 bacteria, and the population after 10 hours is double the population after 1 hour. How many bacteria will there be after 6 hours?
A laptop computer that costs $$\$ 1150$$ new has a book value of $$\$ 550$$ after 2 years. (a) Find the linear model \(V=m t+b\). (b) Find the exponential model \(V=a e^{k t}\) (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.
Use your school's library, the Internet, or some other reference source to write a paper describing John Napier's work with logarithms.
The table shows the time \(t\) (in seconds) required for a car to attain a speed of \(s\) miles per hour from a standing start. $$ \begin{array}{|c|c|} \hline \text { Speed, } s & \text { Time, } t \\ \hline 30 & 3.4 \\ 40 & 5.0 \\ 50 & 7.0 \\ 60 & 9.3 \\ 70 & 12.0 \\ 80 & 15.8 \\ 90 & 20.0 \\ \hline \end{array} $$ Two models for these data are as follows. \(t_{1}=40.757+0.556 s-15.817 \ln s\) \(t_{2}=1.2259+0.0023 s^{2}\) (a) Use the regression feature of a graphing utility to find a linear model \(t_{3}\) and an exponential model \(t_{4}\) for the data. (b) Use a graphing utility to graph the data and each model in the same viewing window. (c) Create a table comparing the data with estimates obtained from each model. (d) Use the results of part (c) to find the sum of the absolute values of the differences between the data and the estimated values given by each model. Based on the four sums, which model do you think best fits the data? Explain.
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