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

## Example Questions

### Example Question #41 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the robomen population due to the transgalactic organic tai cyber conflict is proportional to the population. The population decreased from 6 billion to 450 million between 2063 and 2081. Determine the expected population in 2100.

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 decreased from 6 billion to 450 million between 2063 and 2081, 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 #41 : Constant Of Proportionality

The rate of growth of the culture of E.coli on a piece of room temperature meat is proportional to the population. The population increased from 400 to 1600 between 1:15 and 1:45. Determine the expected population at 3:15.

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 400 to 1600 between 1:15 and 1:45, we can solve for this constant of proportionality. Treat the minutes as decimals after the 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:

Which is why we use refrigerators.

### Example Question #42 : Constant Of Proportionality

The rate of decrease of the population of E. coli in response to a particular antibiotic is proportional to the population. The population decreased from 160,000 to 8,000 between 1:00 and 1:30. Determine the expected population at 1:45.

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 160,000 to 8,000 between 1:00 and 1:30, we can solve for this constant of proportionality. Treat the minutes as decimals after the 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 #44 : How To Find Constant Of Proportionality Of Rate

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:

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:

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

### Example Question #45 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the wolf population in Yosemite is proportional to the population. The population increased from 2500 to 4200 between 2012 and 2015. At what year approximately would the population reach at least 14000?

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 2500 to 4200 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 estimate our time point:

### Example Question #46 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the yeast in a sour dough is proportional to the population. The population increased from 3100 to 11000 in 60 minutes. How many more minutes will it take approximately for the population to reach 25000?

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 3100 to 11000 in 60 minutes, we can solve for this constant of proportionality:

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

### Example Question #47 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the population of salmon of Bodega Bay is proportional to the population. The population increased from 6750 to 9000 between January and March. At what month approximately would the population be 16000?

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 2700 to 9000 between January and March, we can solve for this constant of proportionality. Use the number of the months as they fall in the calendar:

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

### Example Question #48 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the population of hounds of Baskervilles is proportional to the population. The population increased from 13 to 32 between 2013 and 2015. At what year approximately would the population finally exceed 300?

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 13 to 32 between 2013 and 2015, we can solve for this constant of proportionality:

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

### Example Question #43 : Constant Of Proportionality

The rate of growth of the population of kodiak bears in Juno is proportional to the population. The population increased from 1100 to 1300 between 2014 and 2015. At what year approximately would the population be 3000?

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 1100 to 1300 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 estimate our time point:

### Example Question #50 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the population of yeast in some old grape juice is proportional to the population. The population increased from 430 to 1180 after 50 minutes. How many more minutes will it take for the population to exceed 5000?