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

## Example Questions

### Example Question #981 : Rate

The rate of decrease of the small pox incidents due to modern medicine is proportional to the affected population. The population decreased by 89.3 percent between 1990 and 2000. 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:

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

Since the population decreased by 89.3 percent between 1990 and 2000, we can solve for this constant of proportionality:

### Example Question #982 : Rate

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:

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:

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 #983 : Rate

The rate of growth of the number of ants outside of an ice cream shoppe parking lot is proportional to the population. The population increased from 11000 to 21000 between February and June. 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:

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 11000 to 21000 between February and June, 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 #984 : Rate

The rate of growth of the number of moths in  Arizona is proportional to the population. The population increased from 2,100,000 to 4,350,000 between 2013 and 2015. Determine the expected population in 2016.

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:

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 2,100,000 to 4,350,000 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 find an expected population value relative to an initial population value due to the difference in time points:

### Example Question #991 : Rate

The rate of growth of the number of bacterial cells in a dog bowl is proportional to the population. The population increased from 11300 to 28900 between 3:00 and 3:45. Determine the expected population at 5:15.

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:

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 11300 to 28900 between 3:00 and 3:45, we can solve for this constant of proportionality. Treat the minutes as decimals 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 #992 : Rate

The rate of growth of the number of albatrosses plaguing wayward ships is proportional to the population. The population increased from 1123 to 1839 between 1798 and 1802. Determine the expected population in 1834.

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:

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 1123 to 1839 between 1798 and 1802, 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 #97 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the number of living yeast cells as the bread is put into the oven is proportional to the population. The population decreased from 4,500,000,000 to 1,500,000 between 3:15 and 3:45. Determine the expected population at 4:15.

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:

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 4,500,000,000 to 1,500,000 between 3:15 and 3:45, we can solve for this constant of proportionality. Convert minutes to decimals 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 #993 : Rate

The rate of decrease of the number of albino squirrels is proportional to the population. The population decreased from 413 to 136 between 2010 and 2015. Determine the expected population in 2018.

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:

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 413 to 136 between 2010 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 #994 : Rate

The rate of decrease of the number of pollen particles in the air of a room with a vacuum on is proportional to the population. The population decreased from 12345 to 6789 between 3:00 and 4:00. Determine the expected population at 7:00.

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:

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 12345 to 6789 between 3:00 and 4:00, 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 #995 : Rate

The rate of decrease of the number of tunnel snakes is proportional to the population. The population decreased from 2445 to 813 between 2267 and 2269. Determine the expected population in 2277.