Calculus 1 : How to find constant of proportionality of rate

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #181 : Constant Of Proportionality

The rate of change of the number of constant of proportionality calculus problems is proportional to the population. The population increased from 15 to 6750 between September and October. What is the constant of proportionality in months-1?

Possible Answers:

Correct answer:

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 increased from 15 to 6750 between September and October, we can solve for this constant of proportionality (it is useful to treat the months as their number in the calendar):

 

Example Question #182 : Constant Of Proportionality

What is the constant of proportionality of  between  and ?

Possible Answers:

Correct answer:

Explanation:

The constant of proportionality between  and  is given by the equation

In this problem,

 

 

Example Question #181 : Constant Of Proportionality

 of force is required to stretch a spring .  What is the constant of proportionality of the spring?

Possible Answers:

Correct answer:

Explanation:

The relation between the force and stretch of a spring is

 where  is force,  is the spring constant, or proportionality of the spring, and  is how far the spring is strectched.

For this problem

Example Question #184 : Constant Of Proportionality

What is the constant of proportionality of a circle with a diameter of  and a circumference of ?

Possible Answers:

Correct answer:

Explanation:

The relation between circumference and diameter is 

 where  is the circumference of a circle and  is the diameter of the circle.

The constant of proportionality is  for all circles.

Example Question #1081 : Rate

The population of a town grows exponentially from  to  in .  What is the population growth constant?

Possible Answers:

Correct answer:

Explanation:

Exponential growth is modeled by the equation

where  is the final amount,  is the inital amount,  is the growth constant and  is time.

In this problem,  and .  Substituting these variables into the growth equation the solving for  gives us

 

 

Example Question #1082 : Rate

Cobalt-60 has a half-life of .  What is the decay constant of Cobalt-60? 

Possible Answers:

Correct answer:

Explanation:

The half-life of an isotope is the time it takes for half the isotope to disappear.  Isotopes decay exponentially.

Exponential decay is also modeled by the equation 

where  is the final amount,  is the inital amount,  is the growth constant and  is time.

Since half the isotope has disappeared, the final amount  is half the inital amount , or  .

In this problem, .

Substituting these variables into the exponential equation and solving for  gives us

Example Question #181 : Constant Of Proportionality

The number of cats double every .  How many cats will there be after if there are  cats initially?

Possible Answers:

Correct answer:

Explanation:

Exponential growth is modeled by the equation 

where  is the final amount,  is the inital amount,  is the growth constant and  is time.

After , the number of cats has doubled, or the final amount  is double the inital amount , or  .

In this problem, .

Substituting these variables into the exponential equation and solving for  gives us

To find the number cats after  year, , and 

 

 

Example Question #185 : Constant Of Proportionality

The number of students enrolled in college has increased by  every year since .  If   students enrolled in , how many student enrolled in ?

Possible Answers:

Correct answer:

Explanation:

The exponential growth is modeled by the equation 

where  is the final amount,  is the inital amount,  is the growth rate and  is time.

In this problem, , and .  Substituting these values into the equation gives us

 

Example Question #2871 : Functions

The number of CD players owned has decreased by  annually since .  If   people owned CD players in , how many people owned CD players in  ?

Possible Answers:

Correct answer:

Explanation:

The exponential growth is modeled by the equation 

where  is the final amount,  is the inital amount,  is the growth rate and  is time.

In this problem,  and .  because the rate is decreasing.  Substituting these values into the equation gives us

 

Example Question #1086 : Rate

You deposit  into your savings account.  After , your account has  in it.  What is the interest rate of this account if the account was untouched during the ?  

Possible Answers:

Correct answer:

Explanation:

The exponential growth is modeled by the equation 

where  is the final amount,  is the inital amount,  is the growth rate and  is time.

In this problem,  and .  Substituting these variables into the growth equation and solving for r gives us

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