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

## Example Questions

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### Example Question #1 : Exponential Growth Applications

Suppose a blood cell increases proportionally to the present amount.  If there were  blood cells to begin with, and  blood cells are present after  hours, what is the growth constant?

Explanation:

The population size  after some time  is given by:

where  is the initial population.

At the start, there were 30 blood cells.

Substitute this value into the given formula.

After 2 hours, 45 blood cells were present.  Write this in mathematical form.

Substitute this into , and solve for .

### Example Question #1 : How To Find Constant Of Proportionality Of Rate

Given any linear function , determine the direct constant of proportionality

Explanation:

Direct constant of proportionality for any given function y, between any x values, is given by

, where  is the direction constant of proportionality

In the case of a linear function

is the same thing as the slope.

Therefore, the constant of proportionality is

### Example Question #1 : Constant Of Proportionality

Find the direct constant of proportionality of  from  to

Explanation:

To determine the direct constant of proportionality, we determine the rate of change from  and  for .

Rate of change is determined by

.

In our case,  between  and , the rate of change is

.

### Example Question #1 : Constant Of Proportionality

Find the direct constant of proportionality  of  from  to .

Explanation:

Direct constant of proportionality  is given by

.

Since  and

### Example Question #2 : Exponential Growth Applications

Suppose a population of bacteria increases from  to  in . What is the constant of growth?

None of these

Explanation:

The equation for population growth is given by .  is the population,  is the intial value,  is time, and  is the growth constant. We can plug in the values we know at time  and solve for  .

Now that we solved for , we can plug in what we know for time  and solve for .

### Example Question #1 : How To Find Constant Of Proportionality Of Rate

A population of deer grew from 50 to 200 in 7 years. What is the growth constant for this population?

None of these

Explanation:

The equation for population growth is given by . P is the population,  is the intial value,  is time, and  is the growth constant. We can plug in the values we know at time  and solve for  .

Now that we have solved for  we can solve for  at

### Example Question #1 : Constant Of Proportionality

A population of mice has 200 mice. After 6 weeks, there are 1600 mice in the population. What is the constant of growth?

Explanation:

The equation for population growth is given by .  is the population,  is the intial value,  is time, and  is the growth constant. We can plug in the values we know at time  and solve for .

Now that we have  we can solve for  at .

### Example Question #4 : Constant Of Proportionality

Find the direct constant of proportionality  of   from  to

is undefined

Explanation:

Direct constant of proportionality  is given by

, where  is the change in the  position and  is the change in the  position.

Since , and we're going from  to

### Example Question #5 : Exponential Growth Applications

The rate of decrease of the dwindling wolf population of Zion National Park is proportional to the population. The population decreased by 7 percent between 2009 and 2011. What is the constant of proportionality?

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 by 7 percent between 2009 and 2011, we can solve for this constant of proportionality:

### Example Question #2701 : Functions

The rate of growth of the Martian Transgalactic Constituency is proportional to the population. The population increased by 23 percent between 2530 and 2534 AD. What is the constant of proportionality?