Calculus 1 : How to find constant of proportionality of rate

Study concepts, example questions & explanations for Calculus 1

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Example Questions

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:

\displaystyle -0.113

\displaystyle -0.011

\displaystyle -2.416

\displaystyle -2.235

\displaystyle -0.223

Correct answer:

\displaystyle -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:

\displaystyle p(t)=p_0e^{kt}

Where \displaystyle p_0 is an initial population value, and \displaystyle k 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:

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

\displaystyle 0.107=e^{10k}

\displaystyle 10k=ln(0.107)

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

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:

\displaystyle 49000

\displaystyle 55905

\displaystyle 52116

\displaystyle 58344

\displaystyle 49508

Correct answer:

\displaystyle 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:

\displaystyle p(t)=p_0e^{kt}

Where \displaystyle p_0 is an initial population value, \displaystyle t  represents a measure of elapsed time relative to this population value, and \displaystyle k 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:

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

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

\displaystyle 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:

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

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:

\displaystyle 41288

\displaystyle 39503

\displaystyle 43434

\displaystyle 40146

\displaystyle 38102

Correct answer:

\displaystyle 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:

\displaystyle p(t)=p_0e^{kt}

Where \displaystyle p_0 is an initial population value, \displaystyle t  represents a measure of elapsed time relative to this population value, and \displaystyle k 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:

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

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

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

\displaystyle 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:

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

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:

\displaystyle 6341829

\displaystyle 5901671

\displaystyle 6259973

\displaystyle 6609976

\displaystyle 6002819

Correct answer:

\displaystyle 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:

\displaystyle p(t)=p_0e^{kt}

Where \displaystyle p_0 is an initial population value, \displaystyle t  represents a measure of elapsed time relative to this population value, and \displaystyle k 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:

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

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

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

\displaystyle 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:

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

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

The rate of growth of the number of bacterial cells in a dog bowl is proportional to the population. The population increased from 11300 to 28900 between 3:00 and 3:45. Determine the expected population at 5:15.

Possible Answers:

\displaystyle 79984

\displaystyle 112503

\displaystyle 210302

\displaystyle 159665

\displaystyle 189018

Correct answer:

\displaystyle 189018

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:

\displaystyle p(t)=p_0e^{kt}

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

Since the population increased from 11300 to 28900 between 3:00 and 3:45, we can solve for this constant of proportionality. Treat the minutes as decimals by dividing them by 60:

\displaystyle 28900=11300e^{k(3.75-3)}

\displaystyle \frac{289}{113}=e^{0.75k}

\displaystyle 0.75k=ln(\frac{289}{113})

\displaystyle k=\frac{ln(\frac{289}{113})}{0.75}=1.252

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:

\displaystyle P=28900e^{(1.252)(5.25-3.75)} \approx 189018

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

The rate of growth of the number of albatrosses plaguing wayward ships is proportional to the population. The population increased from 1123 to 1839 between 1798 and 1802. Determine the expected population in 1834.

Possible Answers:

\displaystyle 94181

\displaystyle 111545

\displaystyle 86433

\displaystyle 90312

\displaystyle 107224

Correct answer:

\displaystyle 94181

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:

\displaystyle p(t)=p_0e^{kt}

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

Since the population increased from 1123 to 1839 between 1798 and 1802, we can solve for this constant of proportionality:

\displaystyle 1839=1123e^{k(1802-1798)}

\displaystyle \frac{1839}{1123}=e^{4k}

\displaystyle 4k=ln(\frac{1839}{1123})

\displaystyle k=\frac{ln(\frac{1839}{1123})}{4}=0.123

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:

\displaystyle P=1839e^{(0.123)(1834-1802)} \approx 94181

Example Question #991 : Rate

The rate of decrease of the number of living yeast cells as the bread is put into the oven is proportional to the population. The population decreased from 4,500,000,000 to 1,500,000 between 3:15 and 3:45. Determine the expected population at 4:15.

Possible Answers:

\displaystyle 0

\displaystyle 887

\displaystyle 432

\displaystyle 500

\displaystyle 34

Correct answer:

\displaystyle 500

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:

\displaystyle p(t)=p_0e^{kt}

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

Since the population decreased from 4,500,000,000 to 1,500,000 between 3:15 and 3:45, we can solve for this constant of proportionality. Convert minutes to decimals by dividing by 60:

\displaystyle 1500000=4500000000e^{k(3.75-3.25)}

\displaystyle \frac{1}{3000}=e^{0.5k}

\displaystyle 0.5k=ln(\frac{1}{3000})

\displaystyle k=\frac{ln(\frac{1}{3000})}{0.5}=-16.013

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:

\displaystyle P=1500000e^{(-16.013)(4.25-3.75)} \approx 500

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

The rate of decrease of the number of albino squirrels is proportional to the population. The population decreased from 413 to 136 between 2010 and 2015. Determine the expected population in 2018.

Possible Answers:

\displaystyle 265

\displaystyle 70

\displaystyle 112

\displaystyle 23

\displaystyle 89

Correct answer:

\displaystyle 70

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:

\displaystyle p(t)=p_0e^{kt}

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

Since the population decreased from 413 to 136 between 2010 and 2015, we can solve for this constant of proportionality:

\displaystyle 136=413e^{k(2015-2010)}

\displaystyle \frac{136}{413}=e^{5k}

\displaystyle 5k=ln(\frac{136}{413})

\displaystyle k=\frac{ln(\frac{136}{413})}{5}=-0.222

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:

\displaystyle P=136e^{(-0.222)(2018-2015)} \approx 70

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

The rate of decrease of the number of pollen particles in the air of a room with a vacuum on is proportional to the population. The population decreased from 12345 to 6789 between 3:00 and 4:00. Determine the expected population at 7:00.

Possible Answers:

\displaystyle 1129

\displaystyle 1002

\displaystyle 816

\displaystyle 1588

\displaystyle 1234

Correct answer:

\displaystyle 1129

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:

\displaystyle p(t)=p_0e^{kt}

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

Since the population decreased from 12345 to 6789 between 3:00 and 4:00, we can solve for this constant of proportionality:

\displaystyle 6789=12345e^{k(4-3)}

\displaystyle \frac{6789}{12345}=e^{k}

\displaystyle k=ln(\frac{6789}{12345})=-0.598

 

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:

\displaystyle P=6789e^{(-0.598)(7-4)} \approx 1129

Example Question #3812 : Calculus

The rate of decrease of the number of tunnel snakes is proportional to the population. The population decreased from 2445 to 813 between 2267 and 2269. Determine the expected population in 2277.

Possible Answers:

\displaystyle 10

\displaystyle 213

\displaystyle 19

\displaystyle 7

\displaystyle 84

Correct answer:

\displaystyle 10

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:

\displaystyle p(t)=p_0e^{kt}

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

Since the population decreased from 2445 to 813 between 2267 and 2269, we can solve for this constant of proportionality:

\displaystyle 813=2445e^{k(2269-2267)}

\displaystyle \frac{271}{815}=e^{2k}

\displaystyle 2k=ln(\frac{271}{815})

\displaystyle k=\frac{ln(\frac{271}{815})}{2}=-0.551

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:

\displaystyle P=813e^{(-0.551)(2277-2269)} \approx 10

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