Calculus 1 : How to find constant of proportionality of rate

Study concepts, example questions & explanations for Calculus 1

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Example Questions

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

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?

Possible Answers:

Correct answer:

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 #2 : How To Find Constant Of Proportionality Of Rate

Given any linear function , determine the direct constant of proportionality

Possible Answers:

Correct answer:

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 #3 : How To Find Constant Of Proportionality Of Rate

Find the direct constant of proportionality of  from  to

Possible Answers:

Correct answer:

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 #4 : How To Find Constant Of Proportionality Of Rate

Find the direct constant of proportionality  of  from  to .

Possible Answers:

Correct answer:

Explanation:

Direct constant of proportionality  is given by

.

Since  and 

Example Question #5 : How To Find Constant Of Proportionality Of Rate

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

Possible Answers:

None of these

Correct answer:

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?

Possible Answers:

None of these

Correct answer:

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 #2 : How To Find Constant Of Proportionality Of Rate

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

Possible Answers:

Correct answer:

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 #3 : How To Find Constant Of Proportionality Of Rate

Find the direct constant of proportionality  of   from  to 

Possible Answers:

 is undefined 

Correct answer:

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 #9 : How To Find Constant Of Proportionality Of Rate

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?

Possible Answers:

Correct answer:

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 #10 : How To Find Constant Of Proportionality Of Rate

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?

Possible Answers:

Correct answer:

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 between 2530 and 2534 AD, we can solve for this constant of proportionality:

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