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Chapter 3: Problem 5
Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(4^{2 x-7}=64\) (a) \(x=5\) (b) \(x=2\)
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Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x^{2} e^{2 x}+2 x e^{2 x}=0$$
Use the following information for determining sound intensity. The level of sound \(\boldsymbol{\beta}\), in decibels, with an intensity of \(I\), is given by \(\boldsymbol{\beta}=10 \log \left(I / I_{0}\right),\) where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 65 and 66 , find the level of sound \(\boldsymbol{\beta}\). (a) \(I=10^{-11}\) watt per \(\mathrm{m}^{2}\) (rustle of leaves) (b) \(I=10^{2}\) watt per \(\mathrm{m}^{2}\) (jet at 30 meters) (c) \(I=10^{-4}\) watt per \(\mathrm{m}^{2}\) (door slamming) (d) \(I=10^{-2}\) watt per \(\mathrm{m}^{2}\) (siren at 30 meters)
Rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.
The populations \(P\) (in thousands) of Reno, Nevada from 2000 through 2007 can be modeled by \(P=346.8 e^{k t},\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . In \(2005,\) the population of Reno was about 395,000 . (Source: U.S. Census Bureau) (a) Find the value of \(k\). Is the population increasing or decreasing? Explain. (b) Use the model to find the populations of Reno in 2010 and 2015 . Are the results reasonable? Explain. (c) According to the model, during what year will the population reach \(500,000 ?\)
Carbon 14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree growing today. A piece of ancient charcoal contains only \(15 \%\) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of \({ }^{14} \mathrm{C}\) is 5715 years?
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