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Calculus 1 : Calculus

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Example Questions

Example Question #81 : Constant Of Proportionality

The rate of growth of the number of E. coli in a room-temperature stew is proportional to the population. The population increased from 1000 to 3500 between 3:15 and 3:30. What is the constant of proportionality in hours^-1?

Possible Answers:

2.119

3.806

0.835

0.612

5.011

Correct answer:

5.011

Explanation:

We're told that the rate of growth 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 from 1000 to 3500 between 3:15 and 3:30, we can solve for this constant of proportionality. Mind the units; treat the minutes as decimals of an hour by dividing by 60:

$3500=1000e^{k(3.5-3.25)}$

$\frac{7}{2}=e^{0.25k}$

$0.25k=ln(\frac{7}{2})$

$k=\frac{ln(\frac{7}{2})}{0.25}=5.011$

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Example Question #82 : Constant Of Proportionality

The rate of growth of the numer of maned wolves in a select portion of South American grassland is proportional to the population. The population increased from 1800 to 2300 between 2014 and 2015. What is the constant of proportionality in years^-1?

Possible Answers:

0.020

0.189

0.053

0.245

0.117

Correct answer:

0.245

Explanation:

We're told that the rate of growth 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 from 1800 to 2300 between 2014 and 2015, we can solve for this constant of proportionality:

$2300=1800e^{k(2015-2014)}$

$\frac{23}{18}=e^{k}$

$k=ln(\frac{23}{18})=0.245$

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Example Question #3801 : Calculus

The rate of decrease of the number of anacondas in an encroached portion of river is proportional to the population. The population decreased from 1808 to 1117 between 2013 and 2015. What is the constant of proportionality in years^-1?

Possible Answers:

0.241

-1.175

-0.241

1.175

-0.595

Correct answer:

-0.241

Explanation:

We're told that the rate of decrease 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 1808 to 1117 between 2013 and 2015, we can solve for this constant of proportionality:

$1117=1808e^{k(2015-2013)}$

$\frac{1117}{1808}=e^{2k}$

$2k=ln(\frac{1117}{1808})$

$k=\frac{ln(\frac{1117}{1808})}{2}=-0.241$

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Example Question #3801 : Calculus

The rate of decrease of the number of Tazmanian devils near a growing housing development is proportional to the population. The population decreased from 1800 to 713 between 2014 and 2015. What is the constant of proportionality in years^-1?

Possible Answers:

-1.365

-1.117

-1.008

-1.669

-0.926

Correct answer:

-0.926

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 1800 to 713 between 2014 and 2015, we can solve for this constant of proportionality:

$713=1800e^{k(2015-2014)}$

$\frac{713}{1800}=e^{k}$

$k=ln(\frac{713}{1800})=-0.926$

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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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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #81 : Constant Of Proportionality

Example Question #82 : Constant Of Proportionality

Example Question #3801 : Calculus

Example Question #3801 : Calculus

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

All Calculus 1 Resources

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