# Calculus 1 : How to find constant of proportionality of rate

## Example Questions

### Example Question #21 : How To Find Constant Of Proportionality Of Rate

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:

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:

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

### Example Question #21 : Constant Of Proportionality

The rate of growth of the population of mad donkeys is proportional to the population. The population increased from 1450 to 1965 between 2011 and 2015. What is the expected population in 2019?

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:

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

Since the population increased from 1450 to 1965 between 2011 and 2015, we can solve for this constant of proportionality:

Using this, we can calculate the expected value from 2015 to 2019:

### Example Question #23 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the population of lice in a field is proportional to the population. The population increased from 10987 to 15612 between January and March. What is the expected population in October?

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:

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

Since the population increased from 10987 to 15612 between January and March, we can solve for this constant of proportionality. Represent the months as their number in the calendar:

Using this, we can calculate the expected value from March to October:

### Example Question #21 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the population of fruit flies in a salad bar is proportional to the population. The population increased from 38 to 92 in the course of 60 minutes. What is the expected population after another 60 minutes?

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:

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

Since the population increased from 38 to 92 in the course of 60 minutes, we can solve for this constant of proportionality:

Using this, we can calculate the expected value after another 60 minutes:

### Example Question #24 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the number of fungal cells on a loaf of bread is proportional to the population. The population increased from 112 to 357 between 1:00 and 2:00. What is the expected population at 3:00?

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:

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

Since the population increased from 112 to 357 between 1:00 and 2:00, we can solve for this constant of proportionality:

Using this, we can calculate the expected value from 2:00 to 3:00:

### Example Question #25 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the bacteria in a petri dish is proportional to the population. The population increased from 29 to 167 between 3:00 and 6:00. What is the expected population in 8:30?

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:

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

Since the population increased from 29 to 167 between 3:00 and 6:00, we can solve for this constant of proportionality:

Using this, we can calculate the expected value from 6:00 to 8:30:

### Example Question #1 : Asymptotic And Unbounded Behavior

The rate of decrease due to poaching of the elephants in unprotected Sahara is proportional to the population. The population in one region decreased from 1038 to 817 between 2010 and 2015. What is the expected population in 2017?

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:

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

Since the population decreased from 1038 to 817 between 2010 and 2015, we can solve for this constant of proportionality:

Using this, we can calculate the expected value from 2015 to 2017:

### Example Question #26 : How To Find Constant Of Proportionality Of Rate

The rate of decrease in the population of grey wolves in Tennessee is proportional to the population. The population decreased from 5430  to 3740 between 2008 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:

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

Since the population decreased from 5430  to 3740 between 2008 and 2015, we can solve for this constant of proportionality:

Using this, we can calculate the expected value from :

### Example Question #27 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the population of prokaryotes in a pond is proportional to the population. The population increased from 1130 to 1460 between 3:15 and 4:30. What is the expected population in 6:45?

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:

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

Since the population increased from 1130 to 1460 between 3:15 and 4:30, we can solve for this constant of proportionality. Convert the minutes to decimal values of an hour:

Using this, we can calculate the expected value from 4:30 to 6:45:

### Example Question #28 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the constituency of the hivemind is proportional to the population. The population increased from 10800 to 25200 between April and June. What is the expected population in November, when mankind takes its last stand?

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:

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

Since the population increased from 10800 to 25200 between April and June, we can solve for this constant of proportionality. Use the months' number in the calendar:

Using this, we can calculate the expected value in November from June:

Good luck, humanity.