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Calculus 1 : Rate

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Example Question #1071 : Rate

The rate of change of the number of princes turned frogs is proportional to the population. The population increased from 10 to 3900 between 1564 and 1565. What is the constant of proportionality in years^-1?

Possible Answers:

0.88

3.34

1.96

4.23

5.97

Correct answer:

5.97

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 10 to 3900 between 1564 and 1565, we can solve for this constant of proportionality:

$3900=10e^{k(1565-1564)}$

$390=e^{k}$

k=ln(390)=5.97

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Example Question #1072 : Rate

The rate of change of the number of mold spores is proportional to the population. The population increased from 9 to 65610 between May 9th and 13th. What is the constant of proportionality in days^-1?

Possible Answers:

2.22

0.81

2.93

4.38

1.59

Correct answer:

2.22

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 9 to 65610 between May 9th and 13th, we can solve for this constant of proportionality:

$65610=9e^{k(13-9)}$

$7290=e^{4k}$

4k=ln(7290)

$k=\frac{ln(7290)}{4}=2.22$

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Example Question #1073 : Rate

The rate of change of the number of bacteria is proportional to the population. The population increased from 16 to 2560 between 3:15 and 3:25. What is the constant of proportionality in minutes^-1?

Possible Answers:

2.77

0.09

0.51

4.56

1.00

Correct answer:

0.51

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 16 to 2560 between 3:15 and 3:25, we can solve for this constant of proportionality:

$2560=16e^{k(3:25-3:15)}$

$160=e^{10k}$

10k=ln(160)

$k=\frac{ln(160)}{10}=0.51$

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Example Question #1074 : Rate

The rate of change of the number of witches' familiars is proportional to the population. The population increased from 10 to 300 between October 29th and 30th. What is the constant of proportionality in days^-1?

Possible Answers:

0.43

2.11

1.12

0.91

3.40

Correct answer:

3.40

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 10 to 300 between October 29th and 30th, we can solve for this constant of proportionality:

$300=10e^{k(30-29)}$

$30=e^{k}$

k=ln(30)=3.40

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Example Question #1075 : Rate

The rate of change of the number of black squirrels is proportional to the population. The population increased from 110 to 133100 between 2012 and 2015. What is the constant of proportionality in years^-1?

Possible Answers:

4.44

2.37

6.00

0.22

1.19

Correct answer:

2.37

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 110 to 133100 between 2012 and 2015, we can solve for this constant of proportionality:

$133100=110e^{k(2015-2012)}$

$1210=e^{3k}$

3k=ln(1210)

$k=\frac{ln(1210)}{3}=2.37$

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Example Question #171 : Constant Of Proportionality

The rate of change of the number of flap-flaps is proportional to the population. The population increased from 1400 to 9800 between February and April. What is the constant of proportionality in months^-1?

Possible Answers:

1.25

2.12

0.33

2.48

0.97

Correct answer:

0.97

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 9800 between February and April, we can solve for this constant of proportionality (it would be useful to write the months by their place in the calendar):

$9800=1400e^{k(4-2)}$

$7=e^{2k}$

2k=ln(7)

$k=\frac{ln(7)}{2}=0.97$

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Example Question #181 : Constant Of Proportionality

The rate of change of the number of constant of proportionality calculus problems is proportional to the population. The population increased from 15 to 6750 between September and October. What is the constant of proportionality in months^-1?

Possible Answers:

3.23

1.92

6.11

4.07

5.32

Correct answer:

6.11

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 15 to 6750 between September and October, we can solve for this constant of proportionality (it is useful to treat the months as their number in the calendar):

$6750=15e^{k(10-9)}$

$450=e^{k}$

k=ln(450)=6.11

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Example Question #182 : Constant Of Proportionality

What is the constant of proportionality of s(t)=6ln(4t) between t=3 and t=6 ?

Possible Answers:

c=2ln(24)-2ln(12)

c=6ln(6)

c=6ln(12)

c=6ln(24)

Correct answer:

c=2ln(24)-2ln(12)

Explanation:

The constant of proportionality between t=a and t=b is given by the equation

$c=\frac{\Delta y}{\Delta x}=\frac{s(b)-s(a)}{b-a}$

In this problem,

s(3)=6ln(4*3)=6ln(12)

s(6)=6ln(4*6)=6ln(24)

$c=\frac{s(b)-s(a)}{b-a}=\frac{6ln(24)-6ln(12)}{6-3}=\frac{6ln(24)-6ln(12)}{3}$

c=2ln(24)-2ln(12)

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Example Question #183 : Constant Of Proportionality

132N of force is required to stretch a spring 12m . What is the constant of proportionality of the spring?

Possible Answers:

F=11N/m

F=12N/m

F=132N/m

F=120N/m

Correct answer:

F=11N/m

Explanation:

The relation between the force and stretch of a spring is

F=kx where is force, is the spring constant, or proportionality of the spring, and is how far the spring is strectched.

For this problem

$F=kx\Rightarrow k=\frac{F}{x}=\frac{132}{12}=11N/m$

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Example Question #184 : Constant Of Proportionality

What is the constant of proportionality of a circle with a diameter of 3in and a circumference of 9.42in ?

Possible Answers:

9.42

$\pi$

Correct answer:

$\pi$

Explanation:

The relation between circumference and diameter is

$C=\pi D$ where is the circumference of a circle and is the diameter of the circle.

The constant of proportionality is $p=C/D= \pi$ for all circles.

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Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1071 : Rate

Example Question #1072 : Rate

Example Question #1073 : Rate

Example Question #1074 : Rate

Example Question #1075 : Rate

Example Question #171 : Constant Of Proportionality

Example Question #181 : Constant Of Proportionality

Example Question #182 : Constant Of Proportionality

Example Question #183 : Constant Of Proportionality

Example Question #184 : Constant Of Proportionality

All Calculus 1 Resources

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