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

## 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?

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 ?

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?

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 ?

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?

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?

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?

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 ?

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  ?

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 ?