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Calculus 1 : How to find constant of proportionality of rate

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Example Question #11 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the bacteria in an agar dish is proportional to the population. The population increased by 150 percent between 1:15 and 2:30. What is the constant of proportionality?

Possible Answers:

0.039

0.012

0.005

0.008

0.102

Correct answer:

0.012

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 by 150 percent between 1:15 and 2:30, we can solve for this constant of proportionality:

$(1+1.50)y_0=y_0e^{k(2:30 - 1:15)}$

Dealing in minutes:

$2.50=e^{75k}$

75k=ln(2.50)

$k=\frac{ln(2.50)}{75}=0.012$

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Example Question #12 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the duck population in Wingfield is proportional to the population. The population increased by 15 percent between 2001 and 2008. What is the constant of proportionality?

Possible Answers:

0.01997

0.31056

0.02961

0.00341

Correct answer:

0.01997

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 by 15 percent between 2001 and 2008, we can solve for this constant of proportionality:

$(1+.15)y_0=y_0e^{k(2008-2001)}$

$1.15=e^{7k}$

7k=ln(1.15)

$k=\frac{ln(1.15)}{7}=0.01997$

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Example Question #13 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the panda population is proportional to the population. The population decreased by 12 percent between 1990 and 2001. What is the constant of proportionality?

Possible Answers:

-2.526

-0.012

-0.034

-0.191

-0.891

Correct answer:

-0.012

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 decreased by 12 percent between 1990 and 2001, we can solve for this constant of proportionality:

$(1-0.12)y_0=y_0e^{k(2001-1990)}$

$0.88=e^{11k}$

11k=ln(0.88)

$k=\frac{ln(0.88)}{11}=-0.012$

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Example Question #14 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the salmon population of Yuba is proportional to the population. The population increased by 21 percent over the course of seven years. What is the constant of proportionality?

Possible Answers:

-1.755

0.133

0.289

0.030

0.027

Correct answer:

0.027

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 by 21 percent over the course of seven years, we can solve for this constant of proportionality:

$(1+0.21)y_0=y_0e^{7k}$

$1.21=e^{7k}$

7k=ln(1.21)

$k=\frac{ln(1.21)}{7}=0.027$

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Example Question #11 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the robot army of Cloudweb is proportional to the population. The population increased by 89 percent between 2034 and 2037. What is the constant of proportionality for this terrifying army which threatens man's dominance?

Possible Answers:

1.35

-0.46

0.09

0.21

0.76

Correct answer:

0.21

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 by 89 percent between 2034 and 2037, we can solve for this constant of proportionality:

$(1+0.89)y_0=y_0e^{k(2037-2034)}$

$1.89=e^{3k}$

3k=ln(1.89)

$k=\frac{ln(1.89)}{3}=0.21$

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Example Question #11 : Exponential Growth Applications

The rate of decrease of the number of concert attendees to former teen heartthrob Justice Beaver is proportional to the population. The population decreased by 34 percent between 2013 and 2015. What is the constant of proportionality?

Possible Answers:

-0.40

-0.08

0.15

-1.11

-0.21

Correct answer:

-0.21

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 decreased by 34 percent between 2013 and 2015, we can solve for this constant of proportionality:

$(1-0.34)y_0=y_0e^{k(2015-2013)}$

$0.66=e^{2k}$

2k=ln(0.66)

$k=\frac{ln(0.66)}{2}=-0.21$

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Example Question #16 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the Land of Battlecraft players is proportional to the population. The population increased by 72 percent between February and October of 2015. What is the constant of proportionality?

Possible Answers:

0.093

0.068

0.012

0.118

0.059

Correct answer:

0.068

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 by 72 percent between February and October, we can solve for this constant of proportionality. It'll help to represent the months by their number in the year:

$(1+0.72)y_0=y_0e^{k(10-2)}$

$1.72=e^{8k}$

8k=ln(1.72)

$k=\frac{ln(1.72)}{8}=0.068$

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Example Question #31 : Exponential Functions

The rate of decrease of the gluten-eating demographic of the US is proportional to the population. The population decreased by 8 percent between 2014 and 2015. What is the constant of proportionality?

Possible Answers:

-0.109

-0.194

-0.097

-0.083

-0.177

Correct answer:

-0.083

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 decreased by 8 percent between 2014 and 2015, we can solve for this constant of proportionality:

$(1-0.08)y_0=y_0e^{k(2015-2014)}$

$0.92=e^{k}$

k=ln(0.92)

k=-0.083

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Example Question #13 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the monster population of Quiet Hillock is proportional to the population. The population increased by 145 percent between January and August of 1999. What is the constant of proportionality?

Possible Answers:

0.128

0.094

0.021

0.053

0.077

Correct answer:

0.128

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 by 145 percent between January and August, we can solve for this constant of proportionality. It will be helpful to designate the months by their numbers:

$(1+1.45)y_0=y_0e^{k(8-1)}$

$2.45=e^{7k}$

7k=ln(2.45)

$k=\frac{ln(2.45)}{7}=0.128$

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Example Question #11 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the velociraptor population at Mesozoic Terrace is proportional to the population. The population increased by 49 percent between 2010 and 2015. What is the constant of proportionality?

Possible Answers:

1.01

0.12

0.08

0.56

0.73

Correct answer:

0.08

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 by 49 percent between 2010 and 2015, we can solve for this constant of proportionality:

$(1+0.49)y_0=y_0e^{k(2015-2010)}$

$1.49=e^{5k}$

5k=ln(1.49)

$k=\frac{ln(1.49)}{5}=0.08$

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Calculus 1 : How to find constant of proportionality of rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #11 : How To Find Constant Of Proportionality Of Rate

Example Question #12 : How To Find Constant Of Proportionality Of Rate

Example Question #13 : How To Find Constant Of Proportionality Of Rate

Example Question #14 : How To Find Constant Of Proportionality Of Rate

Example Question #11 : How To Find Constant Of Proportionality Of Rate

Example Question #11 : Exponential Growth Applications

Example Question #16 : How To Find Constant Of Proportionality Of Rate

Example Question #31 : Exponential Functions

Example Question #13 : How To Find Constant Of Proportionality Of Rate

Example Question #11 : How To Find Constant Of Proportionality Of Rate

All Calculus 1 Resources

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