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Calculus 1 : Constant of Proportionality

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Example Question #161 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of vampire (bats) is proportional to the population. The population increased from 70 to 3430 between October 2nd and 8th. What is the constant of proportionality in days^-1?

Possible Answers:

0.933

1.576

0.649

0.584

0.209

Correct answer:

0.649

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 from 70 to 3430 between October 2nd and 8th, we can solve for this constant of proportionality:

$3430=70e^{k(8-2)}$

$49=e^{6k}$

6k=ln(49)

$k=\frac{ln(49)}{6}=0.649$

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Example Question #162 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of werewolves is proportional to the population. The population increased from 25 to 625 between October 3rd and 4th. What is the constant of proportionality in days^-1?

Possible Answers:

3.125

3.044

3.219

3.496

3.316

Correct answer:

3.219

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 from 25 to 625 between October 3rd and 4th, we can solve for this constant of proportionality:

$625=25e^{k(4-3)}$

$25=e^{k}$

k=ln(25)=3.219

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Example Question #163 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of friendly ghosts is proportional to the population. The population increased (desparingly) from 20 to 20000 between October 18th and 21st. What is the constant of proportionality in days^-1?

Possible Answers:

1.861

1.293

2.717

1.455

2.303

Correct answer:

2.303

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 (horrifyingly) from 20 to 20000 between October 18th and 21st, we can solve for this constant of proportionality:

$20000=20e^{k(21-18)}$

$1000=e^{3k}$

3k=ln(1000)

$k=\frac{ln(1000)}{3}=2.303$

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Example Question #164 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of talking black cats is proportional to the population. The population increased from 13 to 260 between October 27th and 29th. What is the constant of proportionality in days^-1?

Possible Answers:

1.991

0.217

1.576

1.498

0.716

Correct answer:

1.498

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 from 13 to 260 between October 27th and 29th, we can solve for this constant of proportionality:

$260=13e^{k(29-27)}$

$20=e^{2k}$

2k=ln(20)

$k=\frac{ln(20)}{2}=1.498$

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Example Question #165 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of vampires (not bats) is proportional to the population. The population decreased from 8000 to 250 between November 1st and 5th. What is the constant of proportionality in days^-1?

Possible Answers:

-0.115

-0.866

-0.622

-1.095

-0.467

Correct answer:

-0.866

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 8000 to 250 between November 1st and 5th, we can solve for this constant of proportionality:

$250=8000e^{k(5-1)}$

$\frac{1}{32}=e^{4k}$

$4k=ln(\frac{1}{32})$

$k=\frac{ln(\frac{1}{32})}{4}=-0.866$

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Example Question #166 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of werewolves is proportional to the population. The population decreased from 600 to 30 between November 1st and 15th. What is the constant of proportionality in days^-1?

Possible Answers:

-0.273

-0.214

-0.121

-0.199

-0.236

Correct answer:

-0.214

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 600 to 30 between November 1st and 15th, we can solve for this constant of proportionality:

$30=600e^{k(15-1)}$

$\frac{1}{20}=e^{14k}$

$14k=ln(\frac{1}{20})$

$k=\frac{ln(\frac{1}{20})}{14}=-0.214$

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Example Question #167 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of friendly ghosts is proportional to the population. The population decreased from 10000 to 100 between November 28th and 30th due to the pressures of holiday spending. What is the constant of proportionality in days^-1?

Possible Answers:

-1.266

-0.369

-2.303

-1.874

-2.954

Correct answer:

-2.303

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 10000 to 100 between November 28th and 30th, we can solve for this constant of proportionality:

$100=10000e^{k(30-28)}$

$\frac{1}{100}=e^{2k}$

$2k=ln(\frac{1}{100})$

$k=\frac{ln(\frac{1}{100})}{2}=-2.303$

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Example Question #168 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of spooky, scary skeletons is proportional to the population. The population decreased from 1800 to 30 between November 1st and 2nd. What is the constant of proportionality in days^-1?

Possible Answers:

-4.094

-2.888

-1.028

-3.015

-0.702

Correct answer:

-4.094

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 30 between November 1st and 2nd, we can solve for this constant of proportionality:

$30=1800e^{k(2-1)}$

$\frac{1}{60}=e^{k}$

$k=ln(\frac{1}{60})=-4.094$

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Example Question #169 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of swamp things is proportional to the population. The population decreased from 1900 to 38 between November 2nd and 6th. What is the constant of proportionality in days^-1?

Possible Answers:

-1.447

-0.775

-0.809

-0.978

-1.731

Correct answer:

-0.978

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 1900 to 38 between November 2nd and 6th, we can solve for this constant of proportionality:

$38=1900e^{k(6-2)}$

$\frac{1}{50}=e^{4k}$

$4k=ln(\frac{1}{50})$

$k=\frac{ln(\frac{1}{50})}{4}=-0.978$

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Example Question #2861 : Functions

The rate of change of the number of sycamores is proportional to the population. The population increased from 1400 to 12600 between 2010 and 2012. What is the constant of proportionality in years^-1?

Possible Answers:

1.10

1.99

1.50

4.55

2.34

Correct answer:

1.10

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:

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

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

Since the population increased from 1400 to 12600 between 2010 and 2012, we can solve for this constant of proportionality:

$12600=1400e^{k(2012-2010)}$

$9=e^{2k}$

2k=ln(9)

$k=\frac{ln(9)}{2}=1.10$

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Calculus 1 : Constant of Proportionality

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #161 : How To Find Constant Of Proportionality Of Rate

Example Question #162 : How To Find Constant Of Proportionality Of Rate

Example Question #163 : How To Find Constant Of Proportionality Of Rate

Example Question #164 : How To Find Constant Of Proportionality Of Rate

Example Question #165 : How To Find Constant Of Proportionality Of Rate

Example Question #166 : How To Find Constant Of Proportionality Of Rate

Example Question #167 : How To Find Constant Of Proportionality Of Rate

Example Question #168 : How To Find Constant Of Proportionality Of Rate

Example Question #169 : How To Find Constant Of Proportionality Of Rate

Example Question #2861 : Functions

All Calculus 1 Resources

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