Constant of Proportionality

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AP Calculus AB › Constant of Proportionality

Questions 1 - 10

The rate of growth of the number of cholera causing bacteria in a pound is proportional to the population. The population increased from 100 to 750 between 3:20 and 4:00. What is the constant of proportionality in minutes-1?

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 is an initial population value, and is the constant of proportionality.

Since the population increased from 100 to 750 between 3:20 and 4:00, we can solve for this constant of proportionality:

$750=100e^{k(4:00-3:20)}$

$\frac{15}{2}=e^{40k}$

$40k=ln(\frac{15}{2})$

$k=\frac{ln(\frac{15}{2})}{40}=0.050$

The rate of growth of the number of African wild dogs is proportional to the population. The population increased from 21000 to 35000 between 2013 and 2014. Determine the expected population in 2015.

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 is an initial population value, represents a measure of elapsed time relative to this population value, and is the constant of proportionality.

Since the population increased from 21000 to 35000 between 2013 and 2014, we can solve for this constant of proportionality:

$35000=21000e^{k(2014-2013)}$

$\frac{5}{3}=e^{k}$

$k=ln(\frac{5}{3})=0.511$

Now that the constant of proportionality is known, we can use it to find an expected population value relative to an initial population value due to the difference in time points:

$P=35000e^{(0.511)(2015-2014)} \approx 58344$

The rate of growth of the population of electric mice in Japan is proportional to the population. The population increased from 1800 to 2500 between 2012 and 2015. Determine the expected population in 2018.

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 is an initial population value, represents a measure of elapsed time relative to this population value, and is the constant of proportionality.

Since the population increased from 1800 to 2500 between 2012 and 2015, we can solve for this constant of proportionality:

$2500=1800e^{k(2015-2012)}$

$\frac{25}{18}=e^{3k}$

$3k=ln(\frac{25}{18})$

$k=\frac{ln(\frac{25}{18})}{3}=0.1095$

Now that the constant of proportionality is known, we can use it to find an expected population value relative to an initial population value due to the difference in time points:

$P=2500e^{(2018-2015)(0.1095)} \approx 3472$

The rate of growth of the culture of bacteria on a dirty plate is proportional to the population. The population increased from 50 to 200 between 1:15 and 1:30. At what point in time approximately would the population be 700?

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 is an initial population value, represents a measure of elapsed time relative to this population value, and is the constant of proportionality.

Since the population increased from 50 to 200 between 1:15 and 1:30, we can solve for this constant of proportionality. Treat the minutes as decimals after the hour by dividing by 60:

$200=50e^{k(1.5-1.25)}$

$4=e^{0.25k}$

$k=\frac{ln(4)}{0.25}=5.545$

Now that the constant of proportionality is known, we can use it to estimate our time point:

$700=200e^{5.545(t-1.5)}$

$5.545(t-1.5)=ln(\frac{7}{2})$

$t=1.5+\frac{ln(\frac{7}{2})}{5.545} = 1.726 \rightarrow 1:44$

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?

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 is an initial population value, and is the constant of proportionality.

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

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

$2.13=e^{4k}$

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

The rate of change of the culture size of baceteria on a dirty plate is proportional to the population. The population increased by 32 percent over the course of 35 minutes. What is the constant of proportionality in minutes-1?

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 is an initial population value, and is the constant of proportionality.

Since the population increased by 32 percent over the course of 35 minutes, we can solve for this constant of proportionality:

$(1+0.32)y_0=y_0e^{35k}$

$1.32=e^{35k}$

$k=\frac{ln(1.32)}{35}=0.0079$

The rate of growth of the number of glowing mushrooms in a flooded cave is proportional to the population. The population increased from 110 to 310 between January and July. Determine the expected population in October.

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 is an initial population value, represents a measure of elapsed time relative to this population value, and is the constant of proportionality.

Since the population increased from 110 to 310 between January and July, we can solve for this constant of proportionality. Use the number of the months as they're ordered in the calendar:

$310=110e^{k(7-1)}$

$\frac{31}{11}=e^{6k}$

$6k=ln(\frac{31}{11})$

$k=\frac{ln(\frac{31}{11})}{6}=0.173$

Now that the constant of proportionality is known, we can use it to find an expected population value relative to an initial population value due to the difference in time points:

$P=310e^{(0.173)(10-7)} \approx 521$

The rate of growth of the hogs in Houston is proportional to the population. The population increased from 500 to 900 between 2013 and 2015. What is the expected population in 2020?

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 is an initial population value, and is the constant of proportionality.

Since the population increased from 500 to 900 between 2013 and 2015, we can solve for this constant of proportionality:

$900=500e^{k(2015-2013)}$

$\frac{900}{500}=e^{2k}$

$2k=ln(\frac{900}{500})$

$k=\frac{ln(\frac{900}{500})}{2}=0.294$

Using this, we can calculate the expected value after five years from 2015:

$P=900e^{5(0.294)}\approx 3914$

The rate of change of the number of gila monsters is proportional to the population. The population increased from 1800 to 9000 between 2014 and 2015. What is the constant of proportionality in years-1?

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 is an initial population value, and is the constant of proportionality.

Since the population increased from 1800 to 9000 between 2014 and 2015, we can solve for this constant of proportionality:

$9000=1800e^{k(2015-2014)}$

$5=e^{k}$

The rate of growth of the number of electric eels in Baldwin's Lake is proportional to the population. The population increased by 19.6 percent between 2012 and 2015. What is the constant of proportionality in years-1?

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 is an initial population value, and is the constant of proportionality.

Since the population increased by 19.6 percent between 2012 and 2015, we can solve for this constant of proportionality:

$(1+0.196)y_0=y_0e^{k(2015-2012)}$

$1.196=e^{3k}$

$k=\frac{ln(1.196)}{3}=0.060$

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