{"id":1209,"date":"2021-06-30T17:02:11","date_gmt":"2021-06-30T17:02:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-exponential-growth-and-decay\/"},"modified":"2021-11-17T02:20:09","modified_gmt":"2021-11-17T02:20:09","slug":"problem-set-exponential-growth-and-decay","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-exponential-growth-and-decay\/","title":{"raw":"Problem Set: Exponential Growth and Decay","rendered":"Problem Set: Exponential Growth and Decay"},"content":{"raw":"

True or False<\/em>? If true, prove it. If false, find the true answer (1-4).<\/p>\r\n\r\n

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1.<\/strong> The doubling time for [latex]y={e}^{ct}[\/latex] is [latex](\\text{ln}(2))\\text{\/}(\\text{ln}(c)).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>If you invest [latex]$500,[\/latex] an annual rate of interest of 3% yields more money in the first year than a 2.5% continuous rate of interest.<\/p>\r\n[reveal-answer q=\"fs-id1167793662426\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793662426\"]\r\n

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

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3.\u00a0<\/strong>If you leave a [latex]100\\text{\u00b0}\\text{C}[\/latex] pot of tea at room temperature [latex](25\\text{\u00b0}\\text{C})[\/latex] and an identical pot in the refrigerator [latex](5\\text{\u00b0}\\text{C}),[\/latex] with [latex]k=0.02,[\/latex] the tea in the refrigerator reaches a drinkable temperature [latex](70\\text{\u00b0}\\text{C})[\/latex] more than 5 minutes before the tea at room temperature.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>If given a half-life of [latex]t[\/latex] years, the constant [latex]k[\/latex] for [latex]y={e}^{kt}[\/latex] is calculated by [latex]k=\\text{ln}(1\\text{\/}2)\\text{\/}t.[\/latex]<\/p>\r\n[reveal-answer q=\"15528399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15528399\"]\r\n\r\nFalse; [latex]k=\\frac{\\text{ln}(2)}{t}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (5-, use [latex]y={y}_{0}{e}^{kt}.[\/latex]<\/p>\r\n\r\n

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5.\u00a0<\/strong>If a culture of bacteria doubles in 3 hours, how many hours does it take to multiply by [latex]10?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>If bacteria increase by a factor of 10 in 10 hours, how many hours does it take to increase by [latex]100?[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793286973\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793286973\"]\r\n

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

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7.\u00a0<\/strong>How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is 5730 years.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>If a relic contains 90% as much radiocarbon as new material, can it have come from the time of Christ (approximately 2000 years ago)? Note that the half-life of radiocarbon is 5730 years.<\/p>\r\n[reveal-answer q=\"fs-id1167793312464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793312464\"]\r\n

No. The relic is approximately 871 years old.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.<\/strong> The population of Cairo grew from 5 million to 10 million in 20 years. Use an exponential model to find when the population was 8 million.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>The populations of New York and Los Angeles are growing at 1% and 1.4% a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal?<\/p>\r\n[reveal-answer q=\"fs-id1167794172044\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794172044\"]\r\n

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

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11.\u00a0<\/strong>Suppose the value of [latex]$1[\/latex] in Japanese yen decreases at 2% per year. Starting from [latex]$1=\\text{\u00a5}250,[\/latex] when will [latex]$1=\\text{\u00a5}1?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>The effect of advertising decays exponentially. If 40% of the population remembers a new product after 3 days, how long will 20% remember it?<\/p>\r\n[reveal-answer q=\"fs-id1167793956292\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793956292\"]\r\n

5 days 6 hours 27 minutes<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>If [latex]y=1000[\/latex] at [latex]t=3[\/latex] and [latex]y=3000[\/latex] at [latex]t=4,[\/latex] what was [latex]{y}_{0}[\/latex] at [latex]t=0?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>If [latex]y=100[\/latex] at [latex]t=4[\/latex] and [latex]y=10[\/latex] at [latex]t=8,[\/latex] when does [latex]y=1?[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793286957\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793286957\"]\r\n

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

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15.\u00a0<\/strong>If a bank offers annual interest of 7.5% or continuous interest of [latex]7.25\\text{%},[\/latex] which has a better annual yield?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>What continuous interest rate has the same yield as an annual rate of [latex]9\\text{%}?[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793538337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793538337\"]\r\n

8.618%<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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17.\u00a0<\/strong>If you deposit [latex]$5000[\/latex] at 8% annual interest, how many years can you withdraw [latex]$500[\/latex] (starting after the first year) without running out of money?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>You are trying to save [latex]$50,000[\/latex] in 20 years for college tuition for your child. If interest is a continuous [latex]10\\text{%},[\/latex] how much do you need to invest initially?<\/p>\r\n[reveal-answer q=\"fs-id1167793543297\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793543297\"]\r\n

[latex]$6766.76[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>You are cooling a turkey that was taken out of the oven with an internal temperature of [latex]165\\text{\u00b0}\\text{F}.[\/latex] After 10 minutes of resting the turkey in a [latex]70\\text{\u00b0}\\text{F}[\/latex] apartment, the temperature has reached [latex]155\\text{\u00b0}\\text{F}\\text{.}[\/latex] What is the temperature of the turkey 20 minutes after taking it out of the oven?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>You are trying to thaw some vegetables that are at a temperature of [latex]1\\text{\u00b0}\\text{F}\\text{.}[\/latex] To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of [latex]44\\text{\u00b0}\\text{F}.[\/latex] You check on your vegetables 2 hours after putting them in the refrigerator to find that they are now [latex]12\\text{\u00b0}\\text{F}\\text{.}[\/latex] Plot the resulting temperature curve and use it to determine when the vegetables reach [latex]33\\text{\u00b0}\\text{F}\\text{.}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793541900\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793541900\"]\r\n

9 hours 13 minutes<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era [latex](146[\/latex] million years to 65 million years ago), and you find by radiocarbon dating that there is 0.000001% the amount of radiocarbon. Is this bone from the Cretaceous?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.<\/strong> The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of 24,000 years. If 1 barrel containing 10kg of plutonium-239 is sealed, how many years must pass until only [latex]10g[\/latex] of plutonium-239 is left?<\/p>\r\n[reveal-answer q=\"fs-id1167793477077\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793477077\"]\r\n

239,179 years<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the next set of exercises, use the following table, which features the world population by decade.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Source<\/em>: http:\/\/www.factmonster.com\/ipka\/A0762181.html.<\/caption>\r\n
Years since 1950<\/th>\r\nPopulation (millions)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0<\/td>\r\n2,556<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n3,039<\/td>\r\n<\/tr>\r\n
20<\/td>\r\n3,706<\/td>\r\n<\/tr>\r\n
30<\/td>\r\n4,453<\/td>\r\n<\/tr>\r\n
40<\/td>\r\n5,279<\/td>\r\n<\/tr>\r\n
50<\/td>\r\n6,083<\/td>\r\n<\/tr>\r\n
60<\/td>\r\n6,849<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
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\r\n\r\n23. [T]<\/strong> The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[\/latex] is given by [latex]P(t)=2686{e}^{0.01604t}.[\/latex] Use a graphing calculator to graph the data and the exponential curve together.\r\n\r\n<\/div>\r\n<\/div>\r\n
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\r\n\r\n24. [T]<\/strong> Find and graph the derivative [latex]{y}^{\\prime }[\/latex] of your equation. Where is it increasing and what is the meaning of this increase?\r\n\r\n[reveal-answer q=\"fs-id1167793281540\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793281540\"]\r\n

[latex]P\\prime (t)=43{e}^{0.01604t}.[\/latex] The population is always increasing.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. [T]<\/strong> Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. [T]<\/strong> Find the predicted date when the population reaches 10 billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.<\/p>\r\n[reveal-answer q=\"fs-id1167794005244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794005244\"]\r\n

The population reaches 10 billion people in 2027.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Source<\/em>: http:\/\/www.sfgenealogy.com\/sf\/history\/hgpop.htm.<\/caption>\r\n
Years since 1850<\/strong><\/th>\r\nPopulation (thousands)<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0<\/td>\r\n21.00<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n56.80<\/td>\r\n<\/tr>\r\n
20<\/td>\r\n149.5<\/td>\r\n<\/tr>\r\n
30<\/td>\r\n234.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
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27. [T]<\/strong> The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[\/latex] is given by [latex]P(t)=35.26{e}^{0.06407t}.[\/latex] Use a graphing calculator to graph the data and the exponential curve together.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28. [T]<\/strong> Find and graph the derivative [latex]{y}^{\\prime }[\/latex] of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?<\/p>\r\n[reveal-answer q=\"fs-id1167794127494\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794127494\"]\r\n

[latex]P\\prime (t)=2.259{e}^{0.06407t}.[\/latex] The population is always increasing.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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29. [T]<\/strong> Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?<\/p>\r\n\r\n<\/div>\r\n<\/div>","rendered":"

True or False<\/em>? If true, prove it. If false, find the true answer (1-4).<\/p>\n

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1.<\/strong> The doubling time for [latex]y={e}^{ct}[\/latex] is [latex](\\text{ln}(2))\\text{\/}(\\text{ln}(c)).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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2.\u00a0<\/strong>If you invest [latex]$500,[\/latex] an annual rate of interest of 3% yields more money in the first year than a 2.5% continuous rate of interest.<\/p>\n

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

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3.\u00a0<\/strong>If you leave a [latex]100\\text{\u00b0}\\text{C}[\/latex] pot of tea at room temperature [latex](25\\text{\u00b0}\\text{C})[\/latex] and an identical pot in the refrigerator [latex](5\\text{\u00b0}\\text{C}),[\/latex] with [latex]k=0.02,[\/latex] the tea in the refrigerator reaches a drinkable temperature [latex](70\\text{\u00b0}\\text{C})[\/latex] more than 5 minutes before the tea at room temperature.<\/p>\n<\/div>\n<\/div>\n

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4.\u00a0<\/strong>If given a half-life of [latex]t[\/latex] years, the constant [latex]k[\/latex] for [latex]y={e}^{kt}[\/latex] is calculated by [latex]k=\\text{ln}(1\\text{\/}2)\\text{\/}t.[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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False; [latex]k=\\frac{\\text{ln}(2)}{t}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (5-, use [latex]y={y}_{0}{e}^{kt}.[\/latex]<\/p>\n

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5.\u00a0<\/strong>If a culture of bacteria doubles in 3 hours, how many hours does it take to multiply by [latex]10?[\/latex]<\/p>\n<\/div>\n<\/div>\n

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6.\u00a0<\/strong>If bacteria increase by a factor of 10 in 10 hours, how many hours does it take to increase by [latex]100?[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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20 hours<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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7.\u00a0<\/strong>How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is 5730 years.<\/p>\n<\/div>\n<\/div>\n

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8.\u00a0<\/strong>If a relic contains 90% as much radiocarbon as new material, can it have come from the time of Christ (approximately 2000 years ago)? Note that the half-life of radiocarbon is 5730 years.<\/p>\n

Show Solution<\/span><\/p>\n
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No. The relic is approximately 871 years old.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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9.<\/strong> The population of Cairo grew from 5 million to 10 million in 20 years. Use an exponential model to find when the population was 8 million.<\/p>\n<\/div>\n<\/div>\n

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10.\u00a0<\/strong>The populations of New York and Los Angeles are growing at 1% and 1.4% a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal?<\/p>\n

Show Solution<\/span><\/p>\n
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71.92 years<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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11.\u00a0<\/strong>Suppose the value of [latex]$1[\/latex] in Japanese yen decreases at 2% per year. Starting from [latex]$1=\\text{\u00a5}250,[\/latex] when will [latex]$1=\\text{\u00a5}1?[\/latex]<\/p>\n<\/div>\n<\/div>\n

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12.\u00a0<\/strong>The effect of advertising decays exponentially. If 40% of the population remembers a new product after 3 days, how long will 20% remember it?<\/p>\n

Show Solution<\/span><\/p>\n
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5 days 6 hours 27 minutes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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13.\u00a0<\/strong>If [latex]y=1000[\/latex] at [latex]t=3[\/latex] and [latex]y=3000[\/latex] at [latex]t=4,[\/latex] what was [latex]{y}_{0}[\/latex] at [latex]t=0?[\/latex]<\/p>\n<\/div>\n<\/div>\n

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14.\u00a0<\/strong>If [latex]y=100[\/latex] at [latex]t=4[\/latex] and [latex]y=10[\/latex] at [latex]t=8,[\/latex] when does [latex]y=1?[\/latex]<\/p>\n

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

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15.\u00a0<\/strong>If a bank offers annual interest of 7.5% or continuous interest of [latex]7.25\\text{%},[\/latex] which has a better annual yield?<\/p>\n<\/div>\n<\/div>\n

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16.\u00a0<\/strong>What continuous interest rate has the same yield as an annual rate of [latex]9\\text{%}?[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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8.618%<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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17.\u00a0<\/strong>If you deposit [latex]$5000[\/latex] at 8% annual interest, how many years can you withdraw [latex]$500[\/latex] (starting after the first year) without running out of money?<\/p>\n<\/div>\n<\/div>\n

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18.\u00a0<\/strong>You are trying to save [latex]$50,000[\/latex] in 20 years for college tuition for your child. If interest is a continuous [latex]10\\text{%},[\/latex] how much do you need to invest initially?<\/p>\n

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

[latex]$6766.76[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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19.\u00a0<\/strong>You are cooling a turkey that was taken out of the oven with an internal temperature of [latex]165\\text{\u00b0}\\text{F}.[\/latex] After 10 minutes of resting the turkey in a [latex]70\\text{\u00b0}\\text{F}[\/latex] apartment, the temperature has reached [latex]155\\text{\u00b0}\\text{F}\\text{.}[\/latex] What is the temperature of the turkey 20 minutes after taking it out of the oven?<\/p>\n<\/div>\n<\/div>\n

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20.\u00a0<\/strong>You are trying to thaw some vegetables that are at a temperature of [latex]1\\text{\u00b0}\\text{F}\\text{.}[\/latex] To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of [latex]44\\text{\u00b0}\\text{F}.[\/latex] You check on your vegetables 2 hours after putting them in the refrigerator to find that they are now [latex]12\\text{\u00b0}\\text{F}\\text{.}[\/latex] Plot the resulting temperature curve and use it to determine when the vegetables reach [latex]33\\text{\u00b0}\\text{F}\\text{.}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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9 hours 13 minutes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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21.\u00a0<\/strong>You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era [latex](146[\/latex] million years to 65 million years ago), and you find by radiocarbon dating that there is 0.000001% the amount of radiocarbon. Is this bone from the Cretaceous?<\/p>\n<\/div>\n<\/div>\n

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22.<\/strong> The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of 24,000 years. If 1 barrel containing 10kg of plutonium-239 is sealed, how many years must pass until only [latex]10g[\/latex] of plutonium-239 is left?<\/p>\n

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

239,179 years<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the next set of exercises, use the following table, which features the world population by decade.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
Source<\/em>: http:\/\/www.factmonster.com\/ipka\/A0762181.html.<\/caption>\n
Years since 1950<\/th>\nPopulation (millions)<\/th>\n<\/tr>\n<\/thead>\n
0<\/td>\n2,556<\/td>\n<\/tr>\n
10<\/td>\n3,039<\/td>\n<\/tr>\n
20<\/td>\n3,706<\/td>\n<\/tr>\n
30<\/td>\n4,453<\/td>\n<\/tr>\n
40<\/td>\n5,279<\/td>\n<\/tr>\n
50<\/td>\n6,083<\/td>\n<\/tr>\n
60<\/td>\n6,849<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
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23. [T]<\/strong> The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[\/latex] is given by [latex]P(t)=2686{e}^{0.01604t}.[\/latex] Use a graphing calculator to graph the data and the exponential curve together.<\/p>\n<\/div>\n<\/div>\n

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24. [T]<\/strong> Find and graph the derivative [latex]{y}^{\\prime }[\/latex] of your equation. Where is it increasing and what is the meaning of this increase?<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]P\\prime (t)=43{e}^{0.01604t}.[\/latex] The population is always increasing.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25. [T]<\/strong> Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?<\/p>\n<\/div>\n<\/div>\n

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26. [T]<\/strong> Find the predicted date when the population reaches 10 billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.<\/p>\n

Show Solution<\/span><\/p>\n
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The population reaches 10 billion people in 2027.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century.<\/p>\n\n\n\n\n\n\n\n\n
Source<\/em>: http:\/\/www.sfgenealogy.com\/sf\/history\/hgpop.htm.<\/caption>\n
Years since 1850<\/strong><\/th>\nPopulation (thousands)<\/strong><\/th>\n<\/tr>\n<\/thead>\n
0<\/td>\n21.00<\/td>\n<\/tr>\n
10<\/td>\n56.80<\/td>\n<\/tr>\n
20<\/td>\n149.5<\/td>\n<\/tr>\n
30<\/td>\n234.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
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27. [T]<\/strong> The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[\/latex] is given by [latex]P(t)=35.26{e}^{0.06407t}.[\/latex] Use a graphing calculator to graph the data and the exponential curve together.<\/p>\n<\/div>\n<\/div>\n

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28. [T]<\/strong> Find and graph the derivative [latex]{y}^{\\prime }[\/latex] of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]P\\prime (t)=2.259{e}^{0.06407t}.[\/latex] The population is always increasing.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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29. [T]<\/strong> Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?<\/p>\n<\/div>\n<\/div>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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