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Calculus 1 : Rate

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Example Questions

Example Question #3861 : Calculus

The rate of change of the number of active ghosts is proportional to the population. The population decreased from 20,000 to 10 between 4 AM and 5 AM. What is the constant of proportionality in hours^-1?

Possible Answers:

-3.00

-9.90

-0.69

-7.60

-5.30

Correct answer:

-7.60

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 20,000 to 10 between 4 AM and 5 AM, we can solve for this constant of proportionality:

$10=20000e^{k(5-4)}$

$\frac{1}{2000}=e^{k}$

$k=ln(\frac{1}{2000})=-7.60$

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Example Question #145 : Constant Of Proportionality

The rate of change of the number of skeletons in the skeleton army is proportional to the population. The population increased by 66.6 percent between September and October. What is the constant of proportionality in months^-1?

Possible Answers:

0.510

0.265

0.971

0.030

1.045

Correct answer:

0.510

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 66.6 percent between September and October, we can solve for this constant of proportionality (write the months as their number in the calendar):

$(1+0.666)y_0=y_0e^{k(10-9)}$

$1.666=e^{k}$

k=ln(1.666)=0.510

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Example Question #146 : Constant Of Proportionality

The rate of change of the vampires is proportional to the population. The population increased by 11.9 percent between August and October. What is the constant of proportionality in months^-1?

Possible Answers:

0.531

0.335

0.056

0.155

0.815

Correct answer:

0.056

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 11.9 percent between August and October, we can solve for this constant of proportionality (use the months' numbers in the calendar):

$(1+0.119)y_0=y_0e^{k(10-8)}$

$1.119=e^{2k}$

2k=ln(1.119)

$k=\frac{ln(1.119)}{2}=0.056$

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Example Question #147 : Constant Of Proportionality

The rate of change of the mummies is proportional to the population. The population increased by 123.4 percent between July and October. What is the constant of proportionality in months^-1?

Possible Answers:

0.449

0.047

0.012

0.113

0.268

Correct answer:

0.268

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 123.4 percent between July and October, we can solve for this constant of proportionality (write the months by their number in the calendar):

$(1+1.234)y_0=y_0e^{k(10-7)}$

$2.234=e^{3k}$

3k=ln(2.234)

$k=\frac{ln(2.234)}{3}=0.268$

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Example Question #148 : Constant Of Proportionality

The rate of change of the number of werewolves is proportional to the population. The population increased by 23.4 percent between September and October. What is the constant of proportionality in months^-1?

Possible Answers:

0.089

0.210

0.313

0.101

0.420

Correct answer:

0.210

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 23.4 percent between September and October, we can solve for this constant of proportionality (write the months as their number in the calendar):

$(1+0.234)y_0=y_0e^{k(10-9)}$

$1.234=e^{k}$

k=ln(1.234)=0.210

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Example Question #3866 : Calculus

The rate of change of the number of zombies is proportional to the population. The population increased by 1523 percent between October 1st and October 31st. What is the constant of proportionality in days^-1?

Possible Answers:

0.434

0.578

0.093

0.378

0.188

Correct answer:

0.093

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 1523 percent between October 1st and October 31st, we can solve for this constant of proportionality:

$(1+15.23)y_0=y_0e^{k(31-1)}$

$16.23=e^{30k}$

tk=ln()

$k=\frac{ln(16.23)}{30}=0.093$

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Example Question #2841 : Functions

The rate of change of the number of spooky scary skeletons is proportional to the population. The population increased by 18900 percent between October 15th and October 31st. What is the constant of proportionality in days^-1?

Possible Answers:

1.524

0.152

0.328

0.078

0.897

Correct answer:

0.328

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 18900 percent between October 15th and October 31st, we can solve for this constant of proportionality:

$(1+189)y_0=y_0e^{k(31-15)}$

$190=e^{16k}$

16k=ln(190)

$k=\frac{ln(190)}{16}=0.328$

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Example Question #2842 : Functions

The rate of change of the number of vampires is proportional to the population. The population decreased by 97.5 percent between November 1st and November 8th. What is the constant of proportionality in days^-1?

Possible Answers:

-0.003

-0.255

-1.906

-0.777

-0.527

Correct answer:

-0.527

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 97.5 percent between November 1st and November 8th, we can solve for this constant of proportionality:

$(1-0.975)y_0=y_0e^{k(8-1)}$

$0.025=e^{7k}$

7k=ln(0.025)

$k=\frac{ln(0.025)}{7}=-0.527$

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Example Question #2843 : Functions

The rate of change of the number of werewolves is proportional to the population. The population decreased by 5.3 percent between November 1st and 10th. What is the constant of proportionality in days^-1?

Possible Answers:

-2.099

-0.056

-0.006

-0.111

-0.471

Correct answer:

-0.006

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 5.3 percent between November 1st and 10th, we can solve for this constant of proportionality:

$(1-0.053)y_0=y_0e^{k(10-1)}$

$0.947=e^{9k}$

9k=ln(0.947)

$k=\frac{ln(0.947)}{9}=-0.006$

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Example Question #2844 : Functions

The rate of change of the number of zombies is proportional to the population. The population decreased by 52.5 percent between November 3rd and 6th. What is the constant of proportionality in days^-1?

Possible Answers:

-0.248

-0.089

-0.599

-0.364

-0.672

Correct answer:

-0.248

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 52.5 percent between November 3rd and 6th, we can solve for this constant of proportionality:

$(1-0.525)y_0=y_0e^{k(6-3)}$

$0.475=e^{3k}$

3k=ln(0.475)

$k=\frac{ln(0.475)}{3}=-0.248$

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Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #3861 : Calculus

Example Question #145 : Constant Of Proportionality

Example Question #146 : Constant Of Proportionality

Example Question #147 : Constant Of Proportionality

Example Question #148 : Constant Of Proportionality

Example Question #3866 : Calculus

Example Question #2841 : Functions

Example Question #2842 : Functions

Example Question #2843 : Functions

Example Question #2844 : Functions

All Calculus 1 Resources

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