{"id":2352,"date":"2019-05-13T12:40:18","date_gmt":"2019-05-13T12:40:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2352"},"modified":"2019-05-13T12:41:13","modified_gmt":"2019-05-13T12:41:13","slug":"2-7-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/2-7-section-exercises\/","title":{"raw":"2.7 Section Exercises","rendered":"2.7 Section Exercises"},"content":{"raw":"
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2.7 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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\r\n\r\n1. With what kind of exponential model would half-life<\/em> be associated? What role does half-life play in these models?\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135487320\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135487320\"]\r\n

Half-life is a measure of decay and is thus associated with exponential decay models. The half-life of a substance or quantity is the amount of time it takes for half of the initial amount of that substance or quantity to decay.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. What is carbon dating? Why does it work? Give an example in which carbon dating would be useful.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n3. With what kind of exponential model would doubling time<\/em> be associated? What role does doubling time play in these models?\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135177642\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135177642\"]\r\n

Doubling time is a measure of growth and is thus associated with exponential growth models. The doubling time of a substance or quantity is the amount of time it takes for the initial amount of that substance or quantity to double in size.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. Define Newton\u2019s Law of Cooling. Then name at least three real-world situations where Newton\u2019s Law of Cooling would be applied.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n5. What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135500868\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135500868\"]\r\n

An order of magnitude is the nearest power of ten by which a quantity exponentially grows. It is also an approximate position on a logarithmic scale; Sample response: Orders of magnitude are useful when making comparisons between numbers that differ by a great amount. For example, the mass of Saturn is 95 times greater than the mass of Earth. This is the same as saying that the mass of Saturn is about[latex]\\text{ }{10}^{\\text{2}}\\text{ }[\/latex]times, or 2 orders of magnitude<\/em> greater, than the mass of Earth.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Numeric<\/h4>\r\n
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6. The temperature of an object in degrees Fahrenheit after t <\/em>minutes is represented by the equation[latex]\\text{ }T\\left(t\\right)=68{e}^{-0.0174t}+72.\\text{ }[\/latex]To the nearest degree, what is the temperature of the object after one and a half hours?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the logistic growth model[latex]\\text{ }f\\left(x\\right)=\\frac{150}{1+8{e}^{-2x}}.[\/latex]<\/p>\r\n\r\n

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7. Find and interpret[latex]\\text{ }f\\left(0\\right).\\text{ }[\/latex]Round to the nearest tenth.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135540046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135540046\"]\r\n

[latex]f\\left(0\\right)\\approx 16.7;\\text{ }[\/latex]The amount initially present is about 16.7 units.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8. Find and interpret[latex]\\text{ }f\\left(4\\right).\\text{ }[\/latex]Round to the nearest tenth.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. Find the carrying capacity.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134256674\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134256674\"]\r\n

150<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. Graph the model.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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[latex]x[\/latex]<\/strong><\/td>\r\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
\u20132<\/td>\r\n0.694<\/td>\r\n<\/tr>\r\n
\u20131<\/td>\r\n0.833<\/td>\r\n<\/tr>\r\n
0<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n1.2<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n1.44<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n1.728<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n2.074<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n2.488<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165134054878\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134054878\"]\r\n

exponential;[latex]\\text{ }f\\left(x\\right)={1.2}^{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12. Rewrite[latex]\\text{ }f\\left(x\\right)=1.68{\\left(0.65\\right)}^{x}\\text{ }[\/latex]as an exponential equation with base[latex]\\text{ }e\\text{ }[\/latex]to five significant digits.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Technology<\/h4>\r\n

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.<\/p>\r\n\r\n

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\r\n\r\n13.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n4.079<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n5.296<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n6.159<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n6.828<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n7.375<\/td>\r\n<\/tr>\r\n
7<\/td>\r\n7.838<\/td>\r\n<\/tr>\r\n
8<\/td>\r\n8.238<\/td>\r\n<\/tr>\r\n
9<\/td>\r\n8.592<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n8.908<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165137836553\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137836553\"]\r\n

logarithmic<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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[latex]x[\/latex]<\/strong><\/td>\r\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
1<\/td>\r\n2.4<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n2.88<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n3.456<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n4.147<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n4.977<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n5.972<\/td>\r\n<\/tr>\r\n
7<\/td>\r\n7.166<\/td>\r\n<\/tr>\r\n
8<\/td>\r\n8.6<\/td>\r\n<\/tr>\r\n
9<\/td>\r\n10.32<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n12.383<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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[latex]x[\/latex]<\/strong><\/td>\r\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
4<\/td>\r\n9.429<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n9.972<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n10.415<\/td>\r\n<\/tr>\r\n
7<\/td>\r\n10.79<\/td>\r\n<\/tr>\r\n
8<\/td>\r\n11.115<\/td>\r\n<\/tr>\r\n
9<\/td>\r\n11.401<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n11.657<\/td>\r\n<\/tr>\r\n
11<\/td>\r\n11.889<\/td>\r\n<\/tr>\r\n
12<\/td>\r\n12.101<\/td>\r\n<\/tr>\r\n
13<\/td>\r\n12.295<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165134342556\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134342556\"]\r\n

logarithmic<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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[latex]x[\/latex]<\/strong><\/td>\r\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
1.25<\/td>\r\n5.75<\/td>\r\n<\/tr>\r\n
2.25<\/td>\r\n8.75<\/td>\r\n<\/tr>\r\n
3.56<\/td>\r\n12.68<\/td>\r\n<\/tr>\r\n
4.2<\/td>\r\n14.6<\/td>\r\n<\/tr>\r\n
5.65<\/td>\r\n18.95<\/td>\r\n<\/tr>\r\n
6.75<\/td>\r\n22.25<\/td>\r\n<\/tr>\r\n
7.25<\/td>\r\n23.75<\/td>\r\n<\/tr>\r\n
8.6<\/td>\r\n27.8<\/td>\r\n<\/tr>\r\n
9.25<\/td>\r\n29.75<\/td>\r\n<\/tr>\r\n
10.5<\/td>\r\n33.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in[latex]\\text{ }t\\text{ }[\/latex]years is modeled by the equation[latex]\\text{ }P\\left(t\\right)=\\frac{1000}{1+9{e}^{-0.6t}}.[\/latex]<\/p>\r\n\r\n

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\r\n\r\n17. Graph the function.\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165137611549\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137611549\"]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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18. What is the initial population of fish?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19. To the nearest tenth, what is the doubling time for the fish population?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134275379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134275379\"]\r\n

about[latex]\\text{ }1.4\\text{ }[\/latex]years<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20. To the nearest whole number, what will the fish population be after[latex]\\text{ }2\\text{ }[\/latex]years?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. To the nearest tenth, how long will it take for the population to reach[latex]\\text{ }900?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137647572\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137647572\"]\r\n

about[latex]\\text{ }7.3\\text{ }[\/latex]years<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22. What is the carrying capacity for the fish population? Justify your answer using the graph of[latex]\\text{ }P.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Extensions<\/h4>\r\n
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23. A substance has a half-life of 2.045 minutes. If the initial amount of the substance was 132.8 grams, how many half-lives will have passed before the substance decays to 8.3 grams? What is the total time of decay?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137589542\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137589542\"]\r\n

[latex]4\\text{ }[\/latex]half-lives;[latex]\\text{ }8.18\\text{ }[\/latex]minutes<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24. The formula for an increasing population is given by[latex]\\text{ }P\\left(t\\right)={P}_{0}{e}^{rt}\\text{ }[\/latex]where[latex]\\text{ }{P}_{0}\\text{ }[\/latex]is the initial population and[latex]\\text{ }r>0.\\text{ }[\/latex]Derive a general formula for the time t<\/em> it takes for the population to increase by a factor of M<\/em>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. Recall the formula for calculating the magnitude of an earthquake,[latex]\\text{ }M=\\frac{2}{3}\\mathrm{log}\\left(\\frac{S}{{S}_{0}}\\right).[\/latex] Show each step for solving this equation algebraically for the seismic moment[latex]\\text{ }S.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135208571\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135208571\"]\r\n

[latex]\\begin{array}{l}\\text{ }M=\\frac{2}{3}\\mathrm{log}\\left(\\frac{S}{{S}_{0}}\\right)\\hfill \\\\ \\mathrm{log}\\left(\\frac{S}{{S}_{0}}\\right)=\\frac{3}{2}M\\hfill \\\\ \\text{ }\\frac{S}{{S}_{0}}={10}^{\\frac{3M}{2}}\\hfill \\\\ \\text{ }S={S}_{0}{10}^{\\frac{3M}{2}}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. What is the y<\/em>-intercept of the logistic growth model[latex]\\text{ }y=\\frac{c}{1+a{e}^{-rx}}?\\text{ }[\/latex]Show the steps for calculation. What does this point tell us about the population?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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27. Prove that[latex]\\text{ }{b}^{x}={e}^{x\\mathrm{ln}\\left(b\\right)}\\text{ }[\/latex]for positive[latex]\\text{ }b\\ne 1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135555464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135555464\"]\r\n

Let[latex]\\text{ }y={b}^{x}\\text{ }[\/latex]for some non-negative real number[latex]\\text{ }b\\text{ }[\/latex]such that[latex]\\text{ }b\\ne 1.\\text{ }[\/latex]Then,<\/p>\r\n

[latex]\\begin{array}{l}\\mathrm{ln}\\left(y\\right)=\\mathrm{ln}\\left({b}^{x}\\right)\\hfill \\\\ \\mathrm{ln}\\left(y\\right)=x\\mathrm{ln}\\left(b\\right)\\hfill \\\\ {e}^{\\mathrm{ln}\\left(y\\right)}={e}^{x\\mathrm{ln}\\left(b\\right)}\\hfill \\\\ y={e}^{x\\mathrm{ln}\\left(b\\right)}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Real-World Applications<\/h4>\r\n

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.<\/p>\r\n\r\n

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28. To the nearest hour, what is the half-life of the drug?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n29. Write an exponential model representing the amount of the drug remaining in the patient\u2019s system after[latex]\\text{ }t\\text{ }[\/latex]hours. Then use the formula to find the amount of the drug that would remain in the patient\u2019s system after 3 hours. Round to the nearest milligram.\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135440471\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135440471\"]\r\n

[latex]A=125{e}^{\\left(-0.3567t\\right)};A\\approx 43\\text{ }[\/latex]mg<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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30. Using the model found in the previous exercise, find[latex]\\text{ }f\\left(10\\right)\\text{ }[\/latex]and interpret the result. Round to the nearest hundredth.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use this scenario: A tumor is injected with[latex]\\text{ }0.5\\text{ }[\/latex]grams of Iodine-125, which has a decay rate of[latex]\\text{ }1.15%\\text{ }[\/latex]per day.<\/p>\r\n\r\n

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31. To the nearest day, how long will it take for half of the Iodine-125 to decay?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135532379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135532379\"]\r\n

about[latex]\\text{ }60\\text{ }[\/latex]days<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after[latex]\\text{ }t\\text{ }[\/latex]days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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33. A scientist begins with[latex]\\text{ }\\text{250}\\text{ }[\/latex]grams of a radioactive substance. After[latex]\\text{ }\\text{250}\\text{ }[\/latex]minutes, the sample has decayed to[latex]\\text{ }\\text{32}\\text{ }[\/latex]grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134069337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134069337\"]\r\n

[latex]f\\left(t\\right)=250{e}^{\\left(-0.00914t\\right)};\\text{ }[\/latex]half-life: about[latex]\\text{ }\\text{76}\\text{ }[\/latex]minutes<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n34. The half-life of Radium-226 is[latex]\\text{ }1590\\text{ }[\/latex]years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.\r\n\r\n<\/div>\r\n<\/div>\r\n
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35. The half-life of Erbium-165 is[latex]\\text{ }\\text{10}\\text{.4}\\text{ }[\/latex]hours. What is the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137851455\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137851455\"]\r\n

[latex]r\\approx -0.0667,[\/latex] So the hourly decay rate is about[latex]\\text{ }6.67%[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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36. A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is[latex]\\text{ }\\text{573}0\\text{ }[\/latex]years.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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37. A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was[latex]\\text{ }1350\\text{ }[\/latex]bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after[latex]\\text{ 3 }[\/latex]hours?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135306876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135306876\"]\r\n

[latex]f\\left(t\\right)=1350{e}^{\\left(0.03466t\\right)};\\text{ }[\/latex]after 3 hours:[latex]\\text{ }P\\left(180\\right)\\approx 691,200[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use this scenario: A biologist recorded a count of[latex]\\text{ }360\\text{ }[\/latex]bacteria present in a culture after[latex]\\text{ }5\\text{ }[\/latex]minutes and[latex]\\text{ }1000\\text{ }[\/latex]bacteria present after[latex]\\text{ }20\\text{ }[\/latex]minutes.<\/p>\r\n\r\n

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38. To the nearest whole number, what was the initial population in the culture?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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39. Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137834305\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137834305\"]\r\n

[latex]f\\left(t\\right)=256{e}^{\\left(0.068110t\\right)};\\text{ }[\/latex]doubling time: about[latex]\\text{ }10\\text{ }[\/latex]minutes<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of[latex]\\text{ }\\text{100\u00b0}\\text{ }[\/latex]Fahrenheit was taken off the stove to cool in a[latex]\\text{ }\\text{69\u00b0 F}\\text{ }[\/latex]room. After fifteen minutes, the internal temperature of the soup was[latex]\\text{ }\\text{95\u00b0 F}\\text{.}[\/latex]<\/p>\r\n\r\n

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40. Use Newton\u2019s Law of Cooling to write a formula that models this situation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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41. To the nearest minute, how long will it take the soup to cool to[latex]\\text{ }\\text{80\u00b0 F?}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135536453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135536453\"]\r\n

about[latex]\\text{ }\\text{88}\\text{ }[\/latex]minutes<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. To the nearest degree, what will the temperature be after[latex]\\text{ }2\\text{ }[\/latex]and a half hours?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of[latex]\\text{ }\\text{165\u00b0F}\\text{ }[\/latex] and is allowed to cool in a[latex]\\text{ }\\text{75\u00b0F}\\text{ }[\/latex]room. After half an hour, the internal temperature of the turkey is[latex]\\text{ }\\text{145\u00b0F}\\text{.}[\/latex]<\/p>\r\n\r\n

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43. Write a formula that models this situation.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135503876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135503876\"]\r\n

[latex]T\\left(t\\right)=90{e}^{\\left(-0.008377t\\right)}+75,[\/latex] where[latex]\\text{ }t\\text{ }[\/latex]is in minutes.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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44. To the nearest degree, what will the temperature be after 50 minutes?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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45. To the nearest minute, how long will it take the turkey to cool to[latex]\\text{ }\\text{110\u00b0 F?}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134394590\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134394590\"]\r\n

about[latex]\\text{ }\\text{113}\\text{ }[\/latex]minutes<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.<\/p>\r\n\r\n

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46.\u00a0\"Number<\/span><\/div>\r\n<\/div>\r\n
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47.\u00a0\"Number<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165134085837\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134085837\"]\r\n

[latex]\\mathrm{log}\\left(x\\right)=1.5;\\text{ }x\\approx 31.623[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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48. Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper:[latex]\\text{ }{10}^{-10} \\frac{W}{{m}^{2}},[\/latex]Vacuum:[latex]\\text{ }{10}^{-4}\\frac{W}{{m}^{2}},[\/latex]Jet:[latex]\\text{ }{10}^{2} \\frac{W}{{m}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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49. Recall the formula for calculating the magnitude of an earthquake,[latex]\\text{ }M=\\frac{2}{3}\\mathrm{log}\\left(\\frac{S}{{S}_{0}}\\right).\\text{ }[\/latex]One earthquake has magnitude[latex]\\text{ }\\text{3}.\\text{9}\\text{ }[\/latex]on the MMS scale. If a second earthquake has[latex]\\text{ }\\text{75}0\\text{ }[\/latex]times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135517147\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135517147\"]\r\n

MMS magnitude:[latex]\\text{ }5.82[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use this scenario: The equation[latex]\\text{ }N\\left(t\\right)=\\frac{500}{1+49{e}^{-0.7t}}\\text{ }[\/latex]models the number of people in a town who have heard a rumor after t<\/em> days.<\/p>\r\n\r\n

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50. How many people started the rumor?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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51. To the nearest whole number, how many people will have heard the rumor after 3 days?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134059778\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134059778\"]\r\n

[latex]N\\left(3\\right)\\approx 71[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n52. As[latex]\\text{ }t\\text{ }[\/latex]increases without bound, what value does[latex]\\text{ }N\\left(t\\right)\\text{ }[\/latex]approach? Interpret your answer.\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercise, choose the correct answer choice.<\/p>\r\n\r\n

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53. A doctor and injects a patient with[latex]\\text{ }13\\text{ }[\/latex]milligrams of radioactive dye that decays exponentially. After[latex]\\text{ }12\\text{ }[\/latex]minutes, there are[latex]\\text{ }4.75\\text{ }[\/latex]milligrams of dye remaining in the patient\u2019s system. Which is an appropriate model for this situation?<\/p>\r\n\r\n

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  1. [latex]f\\left(t\\right)=13{\\left(0.0805\\right)}^{t}[\/latex]<\/li>\r\n \t
  2. [latex]f\\left(t\\right)=13{e}^{0.9195t}[\/latex]<\/li>\r\n \t
  3. [latex]f\\left(t\\right)=13{e}^{\\left(-0.0839t\\right)}[\/latex]<\/li>\r\n \t
  4. [latex]f\\left(t\\right)=\\frac{4.75}{1+13{e}^{-0.83925t}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    [reveal-answer q=\"fs-id1165137835581\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137835581\"]\r\n

    C<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

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    2.7 Section Exercises<\/h3>\n
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    Verbal<\/h4>\n
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    1. With what kind of exponential model would half-life<\/em> be associated? What role does half-life play in these models?<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    Half-life is a measure of decay and is thus associated with exponential decay models. The half-life of a substance or quantity is the amount of time it takes for half of the initial amount of that substance or quantity to decay.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    2. What is carbon dating? Why does it work? Give an example in which carbon dating would be useful.<\/p>\n<\/div>\n<\/div>\n

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    3. With what kind of exponential model would doubling time<\/em> be associated? What role does doubling time play in these models?<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    Doubling time is a measure of growth and is thus associated with exponential growth models. The doubling time of a substance or quantity is the amount of time it takes for the initial amount of that substance or quantity to double in size.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    4. Define Newton\u2019s Law of Cooling. Then name at least three real-world situations where Newton\u2019s Law of Cooling would be applied.<\/p>\n<\/div>\n<\/div>\n

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    5. What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    An order of magnitude is the nearest power of ten by which a quantity exponentially grows. It is also an approximate position on a logarithmic scale; Sample response: Orders of magnitude are useful when making comparisons between numbers that differ by a great amount. For example, the mass of Saturn is 95 times greater than the mass of Earth. This is the same as saying that the mass of Saturn is about[latex]\\text{ }{10}^{\\text{2}}\\text{ }[\/latex]times, or 2 orders of magnitude<\/em> greater, than the mass of Earth.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    Numeric<\/h4>\n
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    6. The temperature of an object in degrees Fahrenheit after t <\/em>minutes is represented by the equation[latex]\\text{ }T\\left(t\\right)=68{e}^{-0.0174t}+72.\\text{ }[\/latex]To the nearest degree, what is the temperature of the object after one and a half hours?<\/p>\n<\/div>\n<\/div>\n

    For the following exercises, use the logistic growth model[latex]\\text{ }f\\left(x\\right)=\\frac{150}{1+8{e}^{-2x}}.[\/latex]<\/p>\n

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    7. Find and interpret[latex]\\text{ }f\\left(0\\right).\\text{ }[\/latex]Round to the nearest tenth.<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    [latex]f\\left(0\\right)\\approx 16.7;\\text{ }[\/latex]The amount initially present is about 16.7 units.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    8. Find and interpret[latex]\\text{ }f\\left(4\\right).\\text{ }[\/latex]Round to the nearest tenth.<\/p>\n<\/div>\n<\/div>\n

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    9. Find the carrying capacity.<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    150<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    10. Graph the model.<\/p>\n<\/div>\n<\/div>\n

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    11. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.<\/p>\n<\/div>\n<\/div>\n

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    \n\n\n\n\n\n\n\n\n\n\n\n
    [latex]x[\/latex]<\/strong><\/td>\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n
    \u20132<\/td>\n0.694<\/td>\n<\/tr>\n
    \u20131<\/td>\n0.833<\/td>\n<\/tr>\n
    0<\/td>\n1<\/td>\n<\/tr>\n
    1<\/td>\n1.2<\/td>\n<\/tr>\n
    2<\/td>\n1.44<\/td>\n<\/tr>\n
    3<\/td>\n1.728<\/td>\n<\/tr>\n
    4<\/td>\n2.074<\/td>\n<\/tr>\n
    5<\/td>\n2.488<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
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    Show Solution<\/span><\/p>\n
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    exponential;[latex]\\text{ }f\\left(x\\right)={1.2}^{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    12. Rewrite[latex]\\text{ }f\\left(x\\right)=1.68{\\left(0.65\\right)}^{x}\\text{ }[\/latex]as an exponential equation with base[latex]\\text{ }e\\text{ }[\/latex]to five significant digits.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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    Technology<\/h4>\n

    For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.<\/p>\n

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    13.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
    [latex]x[\/latex]<\/strong><\/td>\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n
    1<\/td>\n2<\/td>\n<\/tr>\n
    2<\/td>\n4.079<\/td>\n<\/tr>\n
    3<\/td>\n5.296<\/td>\n<\/tr>\n
    4<\/td>\n6.159<\/td>\n<\/tr>\n
    5<\/td>\n6.828<\/td>\n<\/tr>\n
    6<\/td>\n7.375<\/td>\n<\/tr>\n
    7<\/td>\n7.838<\/td>\n<\/tr>\n
    8<\/td>\n8.238<\/td>\n<\/tr>\n
    9<\/td>\n8.592<\/td>\n<\/tr>\n
    10<\/td>\n8.908<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
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    Show Solution<\/span><\/p>\n
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    logarithmic<\/p>\n

    \"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    14.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
    [latex]x[\/latex]<\/strong><\/td>\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n
    1<\/td>\n2.4<\/td>\n<\/tr>\n
    2<\/td>\n2.88<\/td>\n<\/tr>\n
    3<\/td>\n3.456<\/td>\n<\/tr>\n
    4<\/td>\n4.147<\/td>\n<\/tr>\n
    5<\/td>\n4.977<\/td>\n<\/tr>\n
    6<\/td>\n5.972<\/td>\n<\/tr>\n
    7<\/td>\n7.166<\/td>\n<\/tr>\n
    8<\/td>\n8.6<\/td>\n<\/tr>\n
    9<\/td>\n10.32<\/td>\n<\/tr>\n
    10<\/td>\n12.383<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
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    15.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
    [latex]x[\/latex]<\/strong><\/td>\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n
    4<\/td>\n9.429<\/td>\n<\/tr>\n
    5<\/td>\n9.972<\/td>\n<\/tr>\n
    6<\/td>\n10.415<\/td>\n<\/tr>\n
    7<\/td>\n10.79<\/td>\n<\/tr>\n
    8<\/td>\n11.115<\/td>\n<\/tr>\n
    9<\/td>\n11.401<\/td>\n<\/tr>\n
    10<\/td>\n11.657<\/td>\n<\/tr>\n
    11<\/td>\n11.889<\/td>\n<\/tr>\n
    12<\/td>\n12.101<\/td>\n<\/tr>\n
    13<\/td>\n12.295<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
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    Show Solution<\/span><\/p>\n
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    logarithmic<\/p>\n

    \"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    16.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
    [latex]x[\/latex]<\/strong><\/td>\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n
    1.25<\/td>\n5.75<\/td>\n<\/tr>\n
    2.25<\/td>\n8.75<\/td>\n<\/tr>\n
    3.56<\/td>\n12.68<\/td>\n<\/tr>\n
    4.2<\/td>\n14.6<\/td>\n<\/tr>\n
    5.65<\/td>\n18.95<\/td>\n<\/tr>\n
    6.75<\/td>\n22.25<\/td>\n<\/tr>\n
    7.25<\/td>\n23.75<\/td>\n<\/tr>\n
    8.6<\/td>\n27.8<\/td>\n<\/tr>\n
    9.25<\/td>\n29.75<\/td>\n<\/tr>\n
    10.5<\/td>\n33.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

    For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in[latex]\\text{ }t\\text{ }[\/latex]years is modeled by the equation[latex]\\text{ }P\\left(t\\right)=\\frac{1000}{1+9{e}^{-0.6t}}.[\/latex]<\/p>\n

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    17. Graph the function.<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
    \"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    18. What is the initial population of fish?<\/p>\n<\/div>\n<\/div>\n

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    19. To the nearest tenth, what is the doubling time for the fish population?<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    about[latex]\\text{ }1.4\\text{ }[\/latex]years<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    20. To the nearest whole number, what will the fish population be after[latex]\\text{ }2\\text{ }[\/latex]years?<\/p>\n<\/div>\n<\/div>\n

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    21. To the nearest tenth, how long will it take for the population to reach[latex]\\text{ }900?[\/latex]<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    about[latex]\\text{ }7.3\\text{ }[\/latex]years<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    22. What is the carrying capacity for the fish population? Justify your answer using the graph of[latex]\\text{ }P.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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    Extensions<\/h4>\n
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    23. A substance has a half-life of 2.045 minutes. If the initial amount of the substance was 132.8 grams, how many half-lives will have passed before the substance decays to 8.3 grams? What is the total time of decay?<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    [latex]4\\text{ }[\/latex]half-lives;[latex]\\text{ }8.18\\text{ }[\/latex]minutes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    24. The formula for an increasing population is given by[latex]\\text{ }P\\left(t\\right)={P}_{0}{e}^{rt}\\text{ }[\/latex]where[latex]\\text{ }{P}_{0}\\text{ }[\/latex]is the initial population and[latex]\\text{ }r>0.\\text{ }[\/latex]Derive a general formula for the time t<\/em> it takes for the population to increase by a factor of M<\/em>.<\/p>\n<\/div>\n<\/div>\n

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    25. Recall the formula for calculating the magnitude of an earthquake,[latex]\\text{ }M=\\frac{2}{3}\\mathrm{log}\\left(\\frac{S}{{S}_{0}}\\right).[\/latex] Show each step for solving this equation algebraically for the seismic moment[latex]\\text{ }S.[\/latex]<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
    \n

    [latex]\\begin{array}{l}\\text{ }M=\\frac{2}{3}\\mathrm{log}\\left(\\frac{S}{{S}_{0}}\\right)\\hfill \\\\ \\mathrm{log}\\left(\\frac{S}{{S}_{0}}\\right)=\\frac{3}{2}M\\hfill \\\\ \\text{ }\\frac{S}{{S}_{0}}={10}^{\\frac{3M}{2}}\\hfill \\\\ \\text{ }S={S}_{0}{10}^{\\frac{3M}{2}}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    26. What is the y<\/em>-intercept of the logistic growth model[latex]\\text{ }y=\\frac{c}{1+a{e}^{-rx}}?\\text{ }[\/latex]Show the steps for calculation. What does this point tell us about the population?<\/p>\n<\/div>\n<\/div>\n

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    27. Prove that[latex]\\text{ }{b}^{x}={e}^{x\\mathrm{ln}\\left(b\\right)}\\text{ }[\/latex]for positive[latex]\\text{ }b\\ne 1.[\/latex]<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    Let[latex]\\text{ }y={b}^{x}\\text{ }[\/latex]for some non-negative real number[latex]\\text{ }b\\text{ }[\/latex]such that[latex]\\text{ }b\\ne 1.\\text{ }[\/latex]Then,<\/p>\n

    [latex]\\begin{array}{l}\\mathrm{ln}\\left(y\\right)=\\mathrm{ln}\\left({b}^{x}\\right)\\hfill \\\\ \\mathrm{ln}\\left(y\\right)=x\\mathrm{ln}\\left(b\\right)\\hfill \\\\ {e}^{\\mathrm{ln}\\left(y\\right)}={e}^{x\\mathrm{ln}\\left(b\\right)}\\hfill \\\\ y={e}^{x\\mathrm{ln}\\left(b\\right)}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    Real-World Applications<\/h4>\n

    For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.<\/p>\n

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    28. To the nearest hour, what is the half-life of the drug?<\/p>\n<\/div>\n<\/div>\n

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    29. Write an exponential model representing the amount of the drug remaining in the patient\u2019s system after[latex]\\text{ }t\\text{ }[\/latex]hours. Then use the formula to find the amount of the drug that would remain in the patient\u2019s system after 3 hours. Round to the nearest milligram.<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
    \n

    [latex]A=125{e}^{\\left(-0.3567t\\right)};A\\approx 43\\text{ }[\/latex]mg<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    30. Using the model found in the previous exercise, find[latex]\\text{ }f\\left(10\\right)\\text{ }[\/latex]and interpret the result. Round to the nearest hundredth.<\/p>\n<\/div>\n<\/div>\n

    For the following exercises, use this scenario: A tumor is injected with[latex]\\text{ }0.5\\text{ }[\/latex]grams of Iodine-125, which has a decay rate of[latex]\\text{ }1.15%\\text{ }[\/latex]per day.<\/p>\n

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    31. To the nearest day, how long will it take for half of the Iodine-125 to decay?<\/p>\n<\/div>\n

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    about[latex]\\text{ }60\\text{ }[\/latex]days<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    32. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after[latex]\\text{ }t\\text{ }[\/latex]days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.<\/p>\n<\/div>\n<\/div>\n

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    33. A scientist begins with[latex]\\text{ }\\text{250}\\text{ }[\/latex]grams of a radioactive substance. After[latex]\\text{ }\\text{250}\\text{ }[\/latex]minutes, the sample has decayed to[latex]\\text{ }\\text{32}\\text{ }[\/latex]grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
    \n

    [latex]f\\left(t\\right)=250{e}^{\\left(-0.00914t\\right)};\\text{ }[\/latex]half-life: about[latex]\\text{ }\\text{76}\\text{ }[\/latex]minutes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    34. The half-life of Radium-226 is[latex]\\text{ }1590\\text{ }[\/latex]years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.<\/p>\n<\/div>\n<\/div>\n

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    35. The half-life of Erbium-165 is[latex]\\text{ }\\text{10}\\text{.4}\\text{ }[\/latex]hours. What is the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    [latex]r\\approx -0.0667,[\/latex] So the hourly decay rate is about[latex]\\text{ }6.67%[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    36. A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is[latex]\\text{ }\\text{573}0\\text{ }[\/latex]years.)<\/p>\n<\/div>\n<\/div>\n

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    37. A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was[latex]\\text{ }1350\\text{ }[\/latex]bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after[latex]\\text{ 3 }[\/latex]hours?<\/p>\n<\/div>\n

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    [latex]f\\left(t\\right)=1350{e}^{\\left(0.03466t\\right)};\\text{ }[\/latex]after 3 hours:[latex]\\text{ }P\\left(180\\right)\\approx 691,200[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    For the following exercises, use this scenario: A biologist recorded a count of[latex]\\text{ }360\\text{ }[\/latex]bacteria present in a culture after[latex]\\text{ }5\\text{ }[\/latex]minutes and[latex]\\text{ }1000\\text{ }[\/latex]bacteria present after[latex]\\text{ }20\\text{ }[\/latex]minutes.<\/p>\n

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    38. To the nearest whole number, what was the initial population in the culture?<\/p>\n<\/div>\n<\/div>\n

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    39. Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?<\/p>\n<\/div>\n

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    [latex]f\\left(t\\right)=256{e}^{\\left(0.068110t\\right)};\\text{ }[\/latex]doubling time: about[latex]\\text{ }10\\text{ }[\/latex]minutes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of[latex]\\text{ }\\text{100\u00b0}\\text{ }[\/latex]Fahrenheit was taken off the stove to cool in a[latex]\\text{ }\\text{69\u00b0 F}\\text{ }[\/latex]room. After fifteen minutes, the internal temperature of the soup was[latex]\\text{ }\\text{95\u00b0 F}\\text{.}[\/latex]<\/p>\n

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    40. Use Newton\u2019s Law of Cooling to write a formula that models this situation.<\/p>\n<\/div>\n<\/div>\n

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    41. To the nearest minute, how long will it take the soup to cool to[latex]\\text{ }\\text{80\u00b0 F?}[\/latex]<\/p>\n<\/div>\n

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    about[latex]\\text{ }\\text{88}\\text{ }[\/latex]minutes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    42. To the nearest degree, what will the temperature be after[latex]\\text{ }2\\text{ }[\/latex]and a half hours?<\/p>\n<\/div>\n<\/div>\n

    For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of[latex]\\text{ }\\text{165\u00b0F}\\text{ }[\/latex] and is allowed to cool in a[latex]\\text{ }\\text{75\u00b0F}\\text{ }[\/latex]room. After half an hour, the internal temperature of the turkey is[latex]\\text{ }\\text{145\u00b0F}\\text{.}[\/latex]<\/p>\n

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    43. Write a formula that models this situation.<\/p>\n<\/div>\n

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    [latex]T\\left(t\\right)=90{e}^{\\left(-0.008377t\\right)}+75,[\/latex] where[latex]\\text{ }t\\text{ }[\/latex]is in minutes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    44. To the nearest degree, what will the temperature be after 50 minutes?<\/p>\n<\/div>\n<\/div>\n

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    45. To the nearest minute, how long will it take the turkey to cool to[latex]\\text{ }\\text{110\u00b0 F?}[\/latex]<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    about[latex]\\text{ }\\text{113}\\text{ }[\/latex]minutes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.<\/p>\n

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    46.\u00a0\"Number<\/span><\/div>\n<\/div>\n
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    47.\u00a0\"Number<\/span><\/div>\n
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    Show Solution<\/span><\/p>\n
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    [latex]\\mathrm{log}\\left(x\\right)=1.5;\\text{ }x\\approx 31.623[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    48. Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper:[latex]\\text{ }{10}^{-10} \\frac{W}{{m}^{2}},[\/latex]Vacuum:[latex]\\text{ }{10}^{-4}\\frac{W}{{m}^{2}},[\/latex]Jet:[latex]\\text{ }{10}^{2} \\frac{W}{{m}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    49. Recall the formula for calculating the magnitude of an earthquake,[latex]\\text{ }M=\\frac{2}{3}\\mathrm{log}\\left(\\frac{S}{{S}_{0}}\\right).\\text{ }[\/latex]One earthquake has magnitude[latex]\\text{ }\\text{3}.\\text{9}\\text{ }[\/latex]on the MMS scale. If a second earthquake has[latex]\\text{ }\\text{75}0\\text{ }[\/latex]times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.<\/p>\n<\/div>\n

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    MMS magnitude:[latex]\\text{ }5.82[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    For the following exercises, use this scenario: The equation[latex]\\text{ }N\\left(t\\right)=\\frac{500}{1+49{e}^{-0.7t}}\\text{ }[\/latex]models the number of people in a town who have heard a rumor after t<\/em> days.<\/p>\n

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    50. How many people started the rumor?<\/p>\n<\/div>\n<\/div>\n

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    51. To the nearest whole number, how many people will have heard the rumor after 3 days?<\/p>\n<\/div>\n

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    Show Solution<\/span><\/p>\n
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    [latex]N\\left(3\\right)\\approx 71[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    52. As[latex]\\text{ }t\\text{ }[\/latex]increases without bound, what value does[latex]\\text{ }N\\left(t\\right)\\text{ }[\/latex]approach? Interpret your answer.<\/p>\n<\/div>\n<\/div>\n

    For the following exercise, choose the correct answer choice.<\/p>\n

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    53. A doctor and injects a patient with[latex]\\text{ }13\\text{ }[\/latex]milligrams of radioactive dye that decays exponentially. After[latex]\\text{ }12\\text{ }[\/latex]minutes, there are[latex]\\text{ }4.75\\text{ }[\/latex]milligrams of dye remaining in the patient\u2019s system. Which is an appropriate model for this situation?<\/p>\n

      \n
    1. [latex]f\\left(t\\right)=13{\\left(0.0805\\right)}^{t}[\/latex]<\/li>\n
    2. [latex]f\\left(t\\right)=13{e}^{0.9195t}[\/latex]<\/li>\n
    3. [latex]f\\left(t\\right)=13{e}^{\\left(-0.0839t\\right)}[\/latex]<\/li>\n
    4. [latex]f\\left(t\\right)=\\frac{4.75}{1+13{e}^{-0.83925t}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
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      Show Solution<\/span><\/p>\n
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      C<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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