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

## Example Questions

### Example Question #31 : Constant Of Proportionality

The rate of decrease of baceterial cells in response to a new antibiotic is proportional to the population. The population decreased from 15800 to 6540 between 3:15 and 4:00. What is the expected population at 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 decreased from 15800 to 6540 between 3:15 and 4:00, we can solve for this constant of proportionality. Treat the minutes as decimal values for the hour:

Using this, we can calculate the expected value from 4:00 to 8:30:

### Example Question #31 : Constant Of Proportionality

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:

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:

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:

### Example Question #33 : Constant Of Proportionality

The rate of growth of the bacteria in a petri dish is proportional to the population. The population increased from 4900 to 7193 between 3:54 and 4:21. Determine the expected population at 5:12.

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

Since the population increased from 4900 to 7193 between 3:54 and 4:21, we can solve for this constant of proportionality. Treat the minutes as decimals of an hour by dividing by 60:

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:

### Example Question #31 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the yeast in a loaf of bread is proportional to the population. The population increased from 178 to 413 between 3:15 and 3:45. Determine the expected population at 5:20.

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

Since the population increased from 178 to 413 between 3:15 and 3:45, we can solve for this constant of proportionality. Treat the minutes as decimals after the hour by dividing them by 60:

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:

### Example Question #931 : Rate

The rate of growth of the population of sentient self-replicating toy soldiers is proportional to the population. The population increased from 12 to 43 between Monday and Wednesday. Determine the expected population on Saturday.

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

Since the population increased from 12 to 43 between Monday and Wednesday, we can solve for this constant of proportionality. Assign the days numbers based on the occurence in the week:

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:

### Example Question #36 : Constant Of Proportionality

The rate of growth of the prokaryotes of Lake Tahoe is proportional to the population. The population increased from 193678 to 254183 between 2014 and 2015. Determine the expected population in 2021.

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

Since the population increased from 193678 to 254183 between 2014 and 2015, we can solve for this constant of proportionality:

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:

### Example Question #37 : Constant Of Proportionality

The rate of growth of the alien spores flooding the atmostphere is proportional to the population. The population increased from 19000 to 58000 between 2013 and 2014. Determine 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,   represents a measure of elapsed time relative to this population value, and  is the constant of proportionality.

Since the population increased from 19000 to 58000 between 2013 and 2014, we can solve for this constant of proportionality:

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:

### Example Question #931 : Rate

The rate of growth of the algae in Bill's unkempt swimming pool is proportional to the population. The population increased from 17902 to 39551 between March and May. Determine the expected population in July.

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

Since the population increased from 17902 to 39551 between March and May, we can solve for this constant of proportionality. Assign the months their number in the calendar to determine time elapsed:

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:

### Example Question #39 : Constant Of Proportionality

The rate of decrease of the population of Bengal tigers has been proportional to the population. The population decreased from 45000 to 1800 between 1900 and 1972. Determine the constant of proportionality for this decrease.

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:

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 decreased from 45000 to 1800 between 1900 and 1972, we can solve for this constant of proportionality:

### Example Question #931 : Rate

The rate of growth of the population of robomen is proportional to the population. The population increased from 100 units to 1500 between 2008 and 2015. Determine the expected population in 2055.

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

Since the population increased from 100 units to 1500 between 2008 and 2015, we can solve for this constant of proportionality:

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:

Better unplug that production facility.