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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 $\displaystyle -3.00$

-9.90 $\displaystyle -9.90$

-0.69 $\displaystyle -0.69$

-7.60 $\displaystyle -7.60$

-5.30 $\displaystyle -5.30$

Correct answer:

-7.60 $\displaystyle -7.60$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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:

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

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

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

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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 $\displaystyle 0.510$

0.265 $\displaystyle 0.265$

0.971 $\displaystyle 0.971$

0.030 $\displaystyle 0.030$

1.045 $\displaystyle 1.045$

Correct answer:

0.510 $\displaystyle 0.510$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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):

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

$1.666=e^{k}$ $\displaystyle 1.666=e^{k}$

$\displaystyle k=ln(1.666)=0.510$

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Example Question #3861 : Calculus

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:

0.056 $\displaystyle 0.056$

0.155 $\displaystyle 0.155$

0.335 $\displaystyle 0.335$

0.531 $\displaystyle 0.531$

0.815 $\displaystyle 0.815$

Correct answer:

0.056 $\displaystyle 0.056$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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):

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

$1.119=e^{2k}$ $\displaystyle 1.119=e^{2k}$

$\displaystyle 2k=ln(1.119)$

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

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Example Question #141 : How To Find Constant Of Proportionality Of Rate

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:

0.268 $\displaystyle 0.268$

0.449 $\displaystyle 0.449$

0.113 $\displaystyle 0.113$

0.047 $\displaystyle 0.047$

0.012 $\displaystyle 0.012$

Correct answer:

0.268 $\displaystyle 0.268$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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):

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

$2.234=e^{3k}$ $\displaystyle 2.234=e^{3k}$

$\displaystyle 3k=ln(2.234)$

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

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Example Question #3863 : Calculus

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:

0.210 $\displaystyle 0.210$

0.101 $\displaystyle 0.101$

0.089 $\displaystyle 0.089$

0.420 $\displaystyle 0.420$

0.313 $\displaystyle 0.313$

Correct answer:

0.210 $\displaystyle 0.210$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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):

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

$1.234=e^{k}$ $\displaystyle 1.234=e^{k}$

$\displaystyle k=ln(1.234)=0.210$

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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 $\displaystyle 0.434$

0.578 $\displaystyle 0.578$

0.093 $\displaystyle 0.093$

0.378 $\displaystyle 0.378$

0.188 $\displaystyle 0.188$

Correct answer:

0.093 $\displaystyle 0.093$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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:

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

$16.23=e^{30k}$ $\displaystyle 16.23=e^{30k}$

tk=ln() $\displaystyle tk=ln()$

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

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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 $\displaystyle 1.524$

0.152 $\displaystyle 0.152$

0.328 $\displaystyle 0.328$

0.078 $\displaystyle 0.078$

0.897 $\displaystyle 0.897$

Correct answer:

0.328 $\displaystyle 0.328$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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:

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

$190=e^{16k}$ $\displaystyle 190=e^{16k}$

$\displaystyle 16k=ln(190)$

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

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Example Question #151 : How To Find Constant Of Proportionality Of Rate

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 $\displaystyle -0.003$

-0.777 $\displaystyle -0.777$

-0.255 $\displaystyle -0.255$

-0.527 $\displaystyle -0.527$

-1.906 $\displaystyle -1.906$

Correct answer:

-0.527 $\displaystyle -0.527$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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:

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

$0.025=e^{7k}$ $\displaystyle 0.025=e^{7k}$

$\displaystyle 7k=ln(0.025)$

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

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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 $\displaystyle -2.099$

-0.056 $\displaystyle -0.056$

-0.006 $\displaystyle -0.006$

-0.111 $\displaystyle -0.111$

-0.471 $\displaystyle -0.471$

Correct answer:

-0.006 $\displaystyle -0.006$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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:

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

$0.947=e^{9k}$ $\displaystyle 0.947=e^{9k}$

$\displaystyle 9k=ln(0.947)$

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

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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 $\displaystyle -0.248$

-0.089 $\displaystyle -0.089$

-0.599 $\displaystyle -0.599$

-0.364 $\displaystyle -0.364$

-0.672 $\displaystyle -0.672$

Correct answer:

-0.248 $\displaystyle -0.248$

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:

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

Where p_0 $\displaystyle p_0$ is an initial population value, and $\displaystyle k$ 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:

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

$0.475=e^{3k}$ $\displaystyle 0.475=e^{3k}$

$\displaystyle 3k=ln(0.475)$

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

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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 #3861 : Calculus

Example Question #141 : How To Find Constant Of Proportionality Of Rate

Example Question #3863 : Calculus

Example Question #3866 : Calculus

Example Question #2841 : Functions

Example Question #151 : How To Find Constant Of Proportionality Of Rate

Example Question #2843 : Functions

Example Question #2844 : Functions

All Calculus 1 Resources

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