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Calculus 1 : Rate

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Example Questions

Example Question #1 : Exponential Growth Applications

Suppose a population of bacteria increases from 100 to 1000 in $30\: min$ . What is the constant of growth?

Possible Answers:

None of these

$\frac{\ln(10)}{30}$

$e^{30}$

$\ln(10)$

Correct answer:

$\frac{\ln(10)}{30}$

Explanation:

The equation for population growth is given by $P=Ae^{kt}$ . is the population, is the intial value, is time, and is the growth constant. We can plug in the values we know at time t=0 and solve for .

$100=Ae^{0k}=A$

Now that we solved for , we can plug in what we know for time t=30 and solve for .

$1000=100e^{30k}$

$e^{30k}=10$

$30k=\ln(10)$

$k=\frac{\ln(10)}{30}$

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Example Question #1 : How To Find Constant Of Proportionality Of Rate

A population of deer grew from 50 to 200 in 7 years. What is the growth constant for this population?

Possible Answers:

$\frac{4}{7}$

$\frac{\ln(4)}{7}$

$\ln(7)$

$\ln(4)$

None of these

Correct answer:

$\frac{\ln(4)}{7}$

Explanation:

The equation for population growth is given by $P=Ae^{kt}$ . P is the population, is the intial value, is time, and is the growth constant. We can plug in the values we know at time t=0 and solve for .

$50=Ae^{0k}=A$

Now that we have solved for we can solve for at t=7

$200=50e^{7k}$

$e^{7k}=4$

$7k=\ln(4)$

$k=\frac{\ln(4)}{7}$

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Example Question #22 : Exponential Functions

A population of mice has 200 mice. After 6 weeks, there are 1600 mice in the population. What is the constant of growth?

Possible Answers:

$\frac{\ln(6)}{8}$

$\ln(8)$

$\frac{\ln(6)}{6}$

$\frac{\ln(8)}{6}$

Correct answer:

$\frac{\ln(8)}{6}$

Explanation:

The equation for population growth is given by $P=Ae^{kt}$ . is the population, is the intial value, is time, and is the growth constant. We can plug in the values we know at time t=0 and solve for .

$200=Ae^{0k}=A$

Now that we have we can solve for at t=6 .

$1600=200e^{6k}$

$e^{6k}=8$

$6k=\ln(8)$

$k=\frac{\ln(8)}{6}$

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Example Question #4 : Constant Of Proportionality

Find the direct constant of proportionality of y=3sec(x) from x=0 to $x=\frac{\pi}{3}$ .

Possible Answers:

$k=\frac{9}{\pi}$

$k=-\frac{\pi}{9}$

$k=\frac{\pi}{9}$

is undefined

Correct answer:

$k=\frac{9}{\pi}$

Explanation:

Direct constant of proportionality is given by

$k=\frac{\Delta y}{\Delta x}$ , where $\Delta y$ is the change in the position and $\Delta x$ is the change in the position.

Since y=3sec(x) , and we're going from x=0 to $x=\frac{\pi}{3}$

$\Delta y=3sec(\frac{\pi}{3})-3sec(0)=6-3=3$

$\Delta x=\frac{\pi}{3}-0=\frac{\pi}{3}$

$k=\frac{3}{\frac{\pi}{3}}=\frac{9}{\pi}$

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Example Question #1 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the dwindling wolf population of Zion National Park is proportional to the population. The population decreased by 7 percent between 2009 and 2011. What is the constant of proportionality?

Possible Answers:

0.130

-0.036

-0.044

-0.011

-0.009

Correct answer:

-0.036

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 7 percent between 2009 and 2011, we can solve for this constant of proportionality:

$(1-0.07)y_0=y_0e^{k(2011-2009)}$

$0.93=e^{2k}$

2k=ln(0.93)

$k=\frac{ln(0.93)}{2}=-0.036$

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Example Question #2701 : Functions

The rate of growth of the Martian Transgalactic Constituency is proportional to the population. The population increased by 23 percent between 2530 and 2534 AD. What is the constant of proportionality?

Possible Answers:

0.05

1.04

-0.37

0.07

0.12

Correct answer:

0.05

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 23 percent between 2530 and 2534 AD, we can solve for this constant of proportionality:

$(1+0.23)y_0=y_0e^{k(2534-2530)}$

$1.23=e^{4k}$

4k=ln(1.23)

$k=\frac{ln(1.23)}{4}=0.05$

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Example Question #2701 : Functions

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

0.102

0.008

0.012

0.005

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 #2 : Exponential Growth Applications

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:

0.31056

0.01997

0.00341

0.02961

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 #3 : Exponential Growth Applications

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:

-0.012

-2.526

-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 #4 : Exponential Growth Applications

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:

0.030

0.289

0.133

-1.755

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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Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 : Exponential Growth Applications

Example Question #1 : How To Find Constant Of Proportionality Of Rate

Example Question #22 : Exponential Functions

Example Question #4 : Constant Of Proportionality

Example Question #1 : How To Find Constant Of Proportionality Of Rate

Example Question #2701 : Functions

Example Question #2701 : Functions

Example Question #2 : Exponential Growth Applications

Example Question #3 : Exponential Growth Applications

Example Question #4 : Exponential Growth Applications

All Calculus 1 Resources

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