Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #63 : How To Find Constant Of Proportionality Of Rate

The rate of change of the trout population in Lake Tahoe is proportional to the population. The population increased by 42 percent between January and April. What is the constant of proportionality in months-1?

Possible Answers:

\displaystyle 0.088

\displaystyle -0.748

\displaystyle -1.036

\displaystyle 0.117

\displaystyle 0.351

Correct answer:

\displaystyle 0.117

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:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population increased by 42 percent between January and April, we can solve for this constant of proportionality. Treat the months as their number in the calendar:

\displaystyle (1+0.42)y_0=y_0e^{k(4-1)}

\displaystyle 1.42=e^{3k}

\displaystyle 3k=ln(1.42)

\displaystyle k=\frac{ln(1.42)}{3}=0.117

Example Question #64 : How To Find Constant Of Proportionality Of Rate

The rate of change of the the number of E. coli in room-temperature hamburger is proportional to the population. The population increased by 100 percent between 4:10 and 4:30. What is the constant of proportionality in seconds-1?

Possible Answers:

\displaystyle 0.6932

\displaystyle 0.0347

\displaystyle 0.0116

\displaystyle 831.7766

\displaystyle 0.0006

Correct answer:

\displaystyle 0.0006

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:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population increased by 100 percent between 4:10 and 4:30, we can solve for this constant of proportionality. Note that units we're asked to use:

\displaystyle (1+1)y_0=y_0e^{k(20*60)}

\displaystyle 2=e^{1200k}

\displaystyle 1200k=ln(2)

\displaystyle k=\frac{ln(2)}{1200}=0.0006

Example Question #65 : How To Find Constant Of Proportionality Of Rate

The rate of change of the number of pike in Lake Mead is proportional to the population. The population increased by 48 percent between 2014 and 2015. What is the constant of proportionality in years-1?

Possible Answers:

\displaystyle 0.392

\displaystyle 0.056

\displaystyle 0.128

\displaystyle 0.204

\displaystyle 0.893

Correct answer:

\displaystyle 0.392

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:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population increased by 48 percent between 2014 and 2015, we can solve for this constant of proportionality:

\displaystyle (1+0.48)y_0=y_0e^{k(2015-2014)}

\displaystyle 1.48=e^{k}

\displaystyle k=ln(1.48)=0.392

 

Example Question #961 : Rate

The rate of growth of the population of wild foxes in Britain is proportional to the population. The population increased by 113 percent between 2011 and 2015. What is the constant of proportionality in years-1?

Possible Answers:

\displaystyle 0.092

\displaystyle 0.283

\displaystyle 0.213

\displaystyle 0.189

\displaystyle 0.240

Correct answer:

\displaystyle 0.189

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:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population increased by 113 percent between 2011 and 2015, we can solve for this constant of proportionality:

\displaystyle (1+1.13)y_0=y_0e^{k(2015-2011)}

\displaystyle 2.13=e^{4k}

\displaystyle 4k=ln(2.13)

\displaystyle k=\frac{ln(2.13)}{4}=0.189

Example Question #61 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the number of swallows in Saksegawa is proportional to the population. The population increased by 8 percent between 2013 and 2015. What is the constant of proportionality in years-1?

Possible Answers:

\displaystyle 0.038

\displaystyle 0.071

\displaystyle 0.027

\displaystyle 0.049

\displaystyle 0.060

Correct answer:

\displaystyle 0.038

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:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population increased by 8 percent between 2013 and 2015, we can solve for this constant of proportionality:

\displaystyle (1+0.08)y_0=y_0e^{k(2015-2013)}

\displaystyle 1.08=e^{2k}

\displaystyle 2k=ln(1.08)

\displaystyle k=\frac{ln(1.08)}{2}=0.038

Example Question #2761 : Functions

The rate of growth of the number of belugas in the pacific is proportional to the population. The population increased by 148 percent between 2010 and 2015. What is the constant of proportionality in years-1?

Possible Answers:

\displaystyle 0.138

\displaystyle 0.182

\displaystyle 0.121

\displaystyle 0.202

\displaystyle 0.165

Correct answer:

\displaystyle 0.182

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:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population increased by 148 percent between 2010 and 2015, we can solve for this constant of proportionality:

\displaystyle (1+1.48)y_0=y_0e^{k(2015-2010)}

\displaystyle 2.48=e^{5k}

\displaystyle 5k=ln(2.48)

\displaystyle k=\frac{ln(2.48)}{5}=0.182

Example Question #71 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the number of grey wolves is proportional to the population. The population decreased by 33 percent between 2008 and 2015. What is the constant of proportionality in years-1?

Possible Answers:

\displaystyle -0.333

\displaystyle -0.103

\displaystyle -0.057

\displaystyle -0.084

\displaystyle -0.019

Correct answer:

\displaystyle -0.057

Explanation:

We're told that the rate of decrease of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population decreased by 33 percent between 2008 and 2015, we can solve for this constant of proportionality:

\displaystyle (1-0.33)y_0=y_0e^{k(2015-2008)}

\displaystyle 0.67=e^{7k}

\displaystyle 7k=ln(0.67)

\displaystyle k=\frac{ln(0.67)}{7}=-0.057

Example Question #962 : Rate

The rate of decrease of the Siberian tiger population of northern Asia is proportional to the population. The population decreased by 18 percent between 2009 and 2015. What is the constant of proportionality in years-1?

Possible Answers:

\displaystyle -0.088

\displaystyle -0.044

\displaystyle -0.033

\displaystyle -0.055

\displaystyle -0.066

Correct answer:

\displaystyle -0.033

Explanation:

We're told that the rate of decrease of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population decreased by 18 percent between 2009 and 2015, we can solve for this constant of proportionality:

\displaystyle (1-0.18)y_0=y_0e^{k(2015-2009)}

\displaystyle 0.82=e^{6k}

\displaystyle 6k=ln(0.82)

\displaystyle k=\frac{ln(0.82)}{6}=-0.033

Example Question #2762 : Functions

The rate of decrease of the number of puffins is proportional to the population. The population decreased by 27 percent between 2003 and 2007. What is the constant of proportionality in years-1?

Possible Answers:

\displaystyle -0.225

\displaystyle -1.701

\displaystyle -0.028

\displaystyle -0.079

\displaystyle -0.011

Correct answer:

\displaystyle -0.079

Explanation:

We're told that the rate of decrease of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population decreased by 27 percent between 2003 and 2007, we can solve for this constant of proportionality:

\displaystyle (1-0.27)y_0=y_0e^{k(2007-2003)}

\displaystyle 0.73=e^{4k}

\displaystyle 4k=ln(0.73)

\displaystyle k=\frac{ln(0.73)}{4}=-0.079

Example Question #72 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the numbers of sunflowers in Nevada is proportional to the population. The population increased from 11000  to 17000 between 2012 and 2015. What is the constant of proportionality in years-1?

Possible Answers:

\displaystyle 0.122

\displaystyle 0.167

\displaystyle 0.112

\displaystyle 0.145

\displaystyle 0.098

Correct answer:

\displaystyle 0.145

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:

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

Where \displaystyle p_0 is an initial population value, and \displaystyle k is the constant of proportionality.

Since the population increased from 11000  to 17000 between 2012 and 2015, we can solve for this constant of proportionality:

\displaystyle 17000=11000e^{k(2015-2012)}

\displaystyle \frac{17}{11}=e^{3k}

\displaystyle 3k=ln(\frac{17}{11})

\displaystyle k=\frac{ln(\frac{17}{11})}{3}=0.145

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