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

## Example Questions

### Example Question #51 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the bacteria on a piece of raw poultry is proportional to the population. The population increased from 1800 to 7200 between 4:15 and 5:00. At what point in time to the nearest minute will the population be 30000?

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 7200 between 4:15 and 5:00, we can solve for this constant of proportionality. Convert the minutes to 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 #52 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the population of wildflowers in a Denver meadow is proportional to the population. The population increased from 1600 to 2000 between 2010 and 2015. At what year approximately will the population reach 6000?

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 1600 to 2000 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 estimate our time point:

### Example Question #53 : How To Find Constant Of Proportionality Of Rate

The rate of growth of the population of jellyfish in the Gulf of Mexico is proportional to the population. The population increased from 12,000 to 32,000 between 2012 and 2015. At what year approximately will the population exceed 600,000?

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 12,000 to 32,000 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 #54 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the number of Staphylococcus in the presence of a potent antibiotic is proportional to the population. The population decreases from 15000 to 4000 over the course of twenty minutes. How many more minutes will it take for the population to be reduced to 10 or less?

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 decreases from 15000 to 4000 over the course of twenty 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 #51 : How To Find Constant Of Proportionality Of Rate

The rate of decrease of the bacteria in the presence of a UV sterilization lamp is proportional to the population. The population on a surgical instrument decreases from 2900 to 500 in the course of an hour. How many more minutes will it take to bring the count under 25?

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 decreases from 2900 to 500 in the course of an hour (or 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 #51 : Constant Of Proportionality

The rate of change of the yeast in a lump of unbaked sour dough is proportional to the population. The population increased by 5 percent over the course of 15 minutes. 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:

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

Since the population  increased by 5 percent over the course of 15 minutes, we can solve for this constant of proportionality:

### Example Question #3771 : Calculus

The rate of change of the culture size of baceteria on a dirty plate is proportional to the population. The population increased by 32 percent over the course of 35 minutes. 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:

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

Since the population increased by 32 percent over the course of 35 minutes, we can solve for this constant of proportionality:

### Example Question #52 : Constant Of Proportionality

The rate of change of the population of Crystal Cove salmon is proportional to the population. The population increased by 23 percent over the course of two years. What is the constant of proportionality in years-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:

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

Since the population increased by 23 percent over the course of two years, we can solve for this constant of proportionality:

### Example Question #53 : Constant Of Proportionality

The rate of change of the Bjorg is proportional to the population. The population increased by 189 percent over the course of 1 year. What is the constant of proportionality in years-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:

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

Since the population increased by 189 percent over the course of 1 year, we can solve for this constant of proportionality:

### Example Question #52 : How To Find Constant Of Proportionality Of Rate

The rate of change of the E. coli number on a piece of raw meat is proportional to the population. The population increased by 6300 percent over the course of 2 hours. What is the constant of proportionality in minutes-1?