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Calculus 1 : Rate

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Example Questions

Example Question #133 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of samoyeds is proportional to the population. The population increased by 269 percent between 2014 and 2015. What is the constant of proportionality in years^-1?

Possible Answers:

2.05

1.31

1.09

1.25

1.73

Correct answer:

1.31

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 269 percent between 2014 and 2015, we can solve for this constant of proportionality:

$(1+2.69)y_0=y_0e^{k(2015-2014)}$

$3.69=e^{k}$

k=ln(3.69)=1.31

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Example Question #134 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of bacteria in a coffee mug is proportional to the population. The population increased by 518 percent between 3:15 and 3:25. What is the constant of proportionality in minutes^-1?

Possible Answers:

0.599

0.182

0.164

2.108

0.115

Correct answer:

0.182

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 518 percent between 3:15 and 3:25, we can solve for this constant of proportionality:

$(1+5.18)y_0=y_0e^{k(3:25-3:15)}$

$6.18=e^{10k}$

10k=ln(6.18)

$k=\frac{ln(6.18)}{10}=0.182$

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Example Question #135 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of emus is proportional to the population. The population decreased by 54 percent between 2013 and 2015. What is the constant of proportionality in years^-1?

Possible Answers:

-0.39

-1.31

-0.31

-1.47

-1.84

Correct answer:

-0.39

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 54 percent between 2013 and 2015, we can solve for this constant of proportionality:

$(1-0.54)y_0=y_0e^{k(2015-2013)}$

$0.46=e^{2k}$

2k=ln(0.46)

$k=\frac{ln(0.46)}{2}=-0.39$

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Example Question #136 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of E. coli cells in the presence of a disinfectant spray is proportional to the population. The population decreased by 99.8 percent between 3:10 and 3:15. What is the constant of proportionality in minutes^-1?

Possible Answers:

-1.24

-4.55

-3.91

-7.82

-6.08

Correct answer:

-1.24

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 99.8 percent between 3:10 and 3:15, we can solve for this constant of proportionality:

$(1-0.998)y_0=y_0e^{k(3:15-3:10)}$

$0.002=e^{5k}$

5k=ln(0.002)

$k=\frac{ln(0.002)}{5}=-1.24$

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Example Question #137 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of sabre cats is proportional to the population. The population decreased by 84.1 percent between 1001 and 1005. What is the constant of proportionality in years^-1?

Possible Answers:

-1.56

-1.84

-3.23

-0.46

-0.04

Correct answer:

-0.46

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 84.1 percent between 1001 and 1005, we can solve for this constant of proportionality:

$(1-0.841)y_0=y_0e^{k(1005-1001)}$

$0.159=e^{4k}$

4k=ln(0.159)

$k=\frac{ln(0.159)}{4}=-0.46$

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Example Question #138 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of body snatchers is proportional to the population. The population increased from 11 to 1331 between 8 am and 9am. What is the constant of proportionality in hours^-1?

Possible Answers:

2.45

7.19

4.80

13.31

9.59

Correct answer:

4.80

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 11 to 1331 between 8 am and 9am, we can solve for this constant of proportionality:

$1331=11e^{k(9-8)}$

$121=e^{k}$

k=ln(121)=4.80

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Example Question #141 : Constant Of Proportionality

The rate of change of the number of man-eating spore plants is proportional to the population. The population increased from 3 to 6561 between May and November. What is the constant of proportionality in months^-1?

Possible Answers:

4.23

0.85

3.92

1.28

5.90

Correct answer:

1.28

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 3 to 6561 between May and November, we can solve for this constant of proportionality. Represent the months by their number in the calendar:

$6561=3e^{k(11-5)}$

$2187=e^{6k}$

6k=ln(2187)

$k=\frac{ln(2187)}{6}=1.28$

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Example Question #142 : Constant Of Proportionality

The rate of change of the number of yeast cells in a pond is proportional to the population. The population increased from 1,800 to 76,000 between 3pm and 8pm. What is the constant of proportionality in hours^-1?

Possible Answers:

2.13

1.98

0.75

1.02

2.46

Correct answer:

0.75

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 1,800 to 76,000 between 3pm and 8pm, we can solve for this constant of proportionality:

$76000=1800e^{k(8-3)}$

$\frac{380}{9}=e^{5k}$

$5k=ln(\frac{380}{9})$

$k=\frac{ln(\frac{380}{9})}{5}=0.75$

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Example Question #143 : Constant Of Proportionality

The rate of change of the number of golems is proportional to the population. The population decreased from 324 to 18 between 897 and 895 BC. What is the constant of proportionality in years^-1?

Possible Answers:

2.75

1.45

-1.45

-3.58

3.58

Correct answer:

-1.45

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 324 to 18 between 897 and 895 BC, we can solve for this constant of proportionality. Remember that BC counts down:

$18=324e^{k(897-895)}$

$\frac{1}{18}=e^{2k}$

$2k=ln(\frac{1}{18})$

$k=\frac{ln(\frac{1}{18})}{2}=-1.45$

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Example Question #144 : Constant Of Proportionality

The rate of change of the number of will-o-wisps is proportional to the population. The population decreased from 9500 to 38 between 3 AM and 6 AM. What is the constant of proportionality in hours^-1?

Possible Answers:

-6.62

-4.42

-1.84

-0.37

-16.56

Correct answer:

-1.84

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 9500 to 38 between 3 AM and 6 AM, we can solve for this constant of proportionality:

$38=9500e^{k(6-3)}$

$\frac{1}{250}=e^{3k}$

$3k=ln(\frac{1}{250})$

$k=\frac{ln(\frac{1}{250})}{3}=-1.84$

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Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #133 : How To Find Constant Of Proportionality Of Rate

Example Question #134 : How To Find Constant Of Proportionality Of Rate

Example Question #135 : How To Find Constant Of Proportionality Of Rate

Example Question #136 : How To Find Constant Of Proportionality Of Rate

Example Question #137 : How To Find Constant Of Proportionality Of Rate

Example Question #138 : How To Find Constant Of Proportionality Of Rate

Example Question #141 : Constant Of Proportionality

Example Question #142 : Constant Of Proportionality

Example Question #143 : Constant Of Proportionality

Example Question #144 : Constant Of Proportionality

All Calculus 1 Resources

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