AP Calculus AB - Rate | Practice Hub

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:

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

$1.196=e^{3k}$

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

Find the rate of change of $y=4x^{3}-12x^{2}+6x+7$ at .

Explanation

To find the rate of change of a polynomial at a point, we must find the first derivative of the polynomial and evaluate the derivative at that point.

For this problem,

$y=4x^{3}-12x^{2}+6x+7$

the first derivative of this expression is

$\frac{dy}{dx}=12x^{2}-24x+6$

at the rate of change is

$\frac{dy}{dx}=12*(2)^{2}-24(2)+6=48-48+6=6$

The rate of change of the number of goblins is proportional to the population. The population increased from 120 to 2400 between October 10th and 13th. What is the constant of proportionality in days-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 120 to 2400 between October 10th and 13th, we can solve for this constant of proportionality:

$2400=120e^{k(13-10)}$

$20=e^{3k}$

$k=\frac{ln(20)}{3}=0.999$

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?

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 66.6 percent between September and October, we can solve for this constant of proportionality (write the months as their number in the calendar):

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

$1.666=e^{k}$

The rate of change of the number of cyborgs on Planet X-038 is proportional to the population. The population increased by 513 percent between 2039 and 2043. 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 513 percent between 2039 and 2043, we can solve for this constant of proportionality:

$(1+5.13)y_0=y_0e^{k(2043-2039)}$

$6.13=e^{4k}$

$k=\frac{ln(6.13)}{4}=0.453$

The rate of change of the number of direwolves is proportional to the population. The population increased by 259 percent between 1605 and 1607. 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 259 percent between 1605 and 1607, we can solve for this constant of proportionality:

$(1+2.59)y_0=y_0e^{k(1607-1605)}$

$3.59=e^{2k}$

$k=\frac{ln(3.59)}{2}=0.64$

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:

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