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Calculus 1 : Rate

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Example Questions

Example Question #81 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the number of roasters at Roasting Man as the event comes to a close is proportional to the population. The population decreased from 51515 to 28907 between Monday and Wednesday. What is the constant of proportionality in days^-1?

Possible Answers:

-0.703

-1.271

-0.115

-1.156

-0.289

Correct answer:

-0.289

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, and is the constant of proportionality.

Since the population decreased from 51515 to 28907 between Monday and Wednesday, we can solve for this constant of proportionality. You may treat the days as their numbered position in a week if that is helpful:

$28907=51515e^{k(4-2)}$

$\frac{28907}{51515}=e^{2k}$

$2k=ln(\frac{28907}{51515})$

$k=\frac{ln(\frac{28907}{51515})}{2}=-0.289$

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Example Question #82 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the number of pigeons during a garbage strike is proportional to the population. The population increased by 27.5 percent between April and June. What is the constant of proportionality in months^-1?

Possible Answers:

0.087

0.192

0.153

0.233

0.121

Correct answer:

0.121

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, and is the constant of proportionality.

Since the population increased by 27.5 percent between April and June, we can solve for this constant of proportionality:

$(1+0.275)y_0=y_0e^{k(6-4)}$

$1.275=e^{2k}$

2k=ln(1.275)

$k=\frac{ln(1.275)}{2}=0.121$

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Example Question #81 : Constant Of Proportionality

The rate of growth of the number of rats in the sewers of Albany is proportional to the population. The population increased by 32.9 percent between 2014 and 2015. What is the constant of proportionality in years^-1?

Possible Answers:

0.376

0.189

0.284

0.333

0.114

Correct answer:

0.284

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, and is the constant of proportionality.

Since the population increased by 32.9 percent between 2014 and 2015, we can solve for this constant of proportionality:

$(1+0.329)y_0=y_0e^{k(2015-2014)}$

$1.329=e^{k}$

k=ln(1.329)=0.284

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Example Question #82 : Constant Of Proportionality

The rate of growth of the number of ants in a hill outside of a candy store is proportional to the population. The population increased by 54.8 percent between March and June. What is the constant of proportionality in years^-1?

Possible Answers:

0.399

0.826

0.146

0.505

1.748

Correct answer:

1.748

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, and is the constant of proportionality.

Since the population increased by 54.8 percent between March and June, we can solve for this constant of proportionality. Note that we're asked for the answer in inverse years:

$(1+0.548)y_0=y_0e^{k(\frac{6-3}{12})}$

$1.548=e^{\frac{1}{4}k}$

$\frac{1}{4}k=ln(1.548)$

$k=\frac{ln(1.548)}{\frac{1}{4}}=1.748$

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Example Question #85 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the number of roaches due to lack of food is proportional to the population. The population decreased by 11.2 percent between 2014 and 2015. What is the constant of proportionality in years^-1?

Possible Answers:

-0.142

-0.199

-0.112

-0.119

-0.035

Correct answer:

-0.119

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, and is the constant of proportionality.

Since the population decreased by 11.2 percent between 2014 and 2015, we can solve for this constant of proportionality:

$(1-0.112)y_0=y_0e^{k(2015-2014)}$

$0.888=e^{k}$

k=ln(0.888)=-0.119

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Example Question #86 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the number of earthworms in a field due to new pesiticides is proportional to the population. The population decreased by 63.5 percent between June and October. What is the constant of proportionality in years^-1?

Possible Answers:

-0.008

-0.999

-0.252

-3.024

-1.178

Correct answer:

-3.024

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, and is the constant of proportionality.

Since the population decreased by 63.5 percent between June and October, we can solve for this constant of proportionality. Recall we're asked for units in terms of years:

$(1-0.635)y_0=y_0e^{k(\frac{10-6}{12})}$

$0.365=e^{\frac{1}{3}k}$

$\frac{1}{3}k=ln(0.365)$

$k=\frac{ln(0.365)}{\frac{1}{3}}=-3.024$

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Example Question #91 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the small pox incidents due to modern medicine is proportional to the affected population. The population decreased by 89.3 percent between 1990 and 2000. What is the constant of proportionality in years^-1?

Possible Answers:

-0.011

-2.235

-0.223

-0.113

-2.416

Correct answer:

-0.223

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, and is the constant of proportionality.

Since the population decreased by 89.3 percent between 1990 and 2000, we can solve for this constant of proportionality:

$(1-0.893)y_0=y_0e^{k(2000-1990)}$

$0.107=e^{10k}$

10k=ln(0.107)

$k=\frac{ln(0.107)}{10}=-0.223$

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Example Question #92 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the number of African wild dogs is proportional to the population. The population increased from 21000 to 35000 between 2013 and 2014. Determine the expected population in 2015.

Possible Answers:

55905

52116

49000

49508

58344

Correct answer:

58344

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, represents a measure of elapsed time relative to this population value, and is the constant of proportionality.

Since the population increased from 21000 to 35000 between 2013 and 2014, we can solve for this constant of proportionality:

$35000=21000e^{k(2014-2013)}$

$\frac{5}{3}=e^{k}$

$k=ln(\frac{5}{3})=0.511$

Now that the constant of proportionality is known, we can use it to find an expected population value relative to an initial population value due to the difference in time points:

$P=35000e^{(0.511)(2015-2014)} \approx 58344$

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Example Question #93 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the number of ants outside of an ice cream shoppe parking lot is proportional to the population. The population increased from 11000 to 21000 between February and June. Determine the expected population in October.

Possible Answers:

43434

39503

38102

40146

41288

Correct answer:

40146

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, represents a measure of elapsed time relative to this population value, and is the constant of proportionality.

Since the population increased from 11000 to 21000 between February and June, we can solve for this constant of proportionality:

$21000=11000e^{k(6-2)}$

$\frac{21}{11}=e^{4k}$

$4k=ln(\frac{21}{11})$

$k=\frac{ln(\frac{21}{11})}{4}=0.162$

Now that the constant of proportionality is known, we can use it to find an expected population value relative to an initial population value due to the difference in time points:

$P=21000e^{(10-6)(0.162)} \approx 40146$

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Example Question #94 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the number of moths in Arizona is proportional to the population. The population increased from 2,100,000 to 4,350,000 between 2013 and 2015. Determine the expected population in 2016.

Possible Answers:

6341829

5901671

6609976

6259973

6002819

Correct answer:

6259973

Explanation:

We're told that the rate of change of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, represents a measure of elapsed time relative to this population value, and is the constant of proportionality.

Since the population increased from 2,100,000 to 4,350,000 between 2013 and 2015, we can solve for this constant of proportionality:

$4350000=2100000e^{k(2015-2013)}$

$\frac{435}{210}=e^{2k}$

$2k=ln(\frac{435}{210})$

$k=\frac{ln(\frac{435}{210})}{2}=0.364$

Now that the constant of proportionality is known, we can use it to find an expected population value relative to an initial population value due to the difference in time points:

$P=4350000e^{(0.364)(2016-2015)} \approx 6259973$

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Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #81 : How To Find Constant Of Proportionality Of Rate

Example Question #82 : How To Find Constant Of Proportionality Of Rate

Example Question #81 : Constant Of Proportionality

Example Question #82 : Constant Of Proportionality

Example Question #85 : How To Find Constant Of Proportionality Of Rate

Example Question #86 : How To Find Constant Of Proportionality Of Rate

Example Question #91 : How To Find Constant Of Proportionality Of Rate

Example Question #92 : How To Find Constant Of Proportionality Of Rate

Example Question #93 : How To Find Constant Of Proportionality Of Rate

Example Question #94 : How To Find Constant Of Proportionality Of Rate

All Calculus 1 Resources

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