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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?
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 #2871 : Functions
What is the constant of proportionality of between and ?
The constant of proportionality between and is given by the equation
In this problem,
Example Question #181 : How To Find Constant Of Proportionality Of Rate
of force is required to stretch a spring . What is the constant of proportionality of the spring?
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 #183 : Constant Of Proportionality
What is the constant of proportionality of a circle with a diameter of and a circumference of ?
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 #2871 : Functions
The population of a town grows exponentially from to in . What is the population growth constant?
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 #185 : Constant Of Proportionality
Cobalt-60 has a half-life of . What is the decay constant of Cobalt-60?
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 #186 : Constant Of Proportionality
The number of cats double every . How many cats will there be after if there are cats initially?
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 #187 : 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 ?
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 #188 : Constant Of Proportionality
The number of CD players owned has decreased by annually since . If people owned CD players in , how many people owned CD players in ?
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 #189 : Constant Of Proportionality
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 ?
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