Q: Suppose you invest $1000 at a continuous interest rate of 4% per year. What will be the value of your investment after 10 years?

$$ 1000 \times e$$^{0.04 \times 10}$$ $$

Q: A population of bacteria decays continuously at a rate of $$ r = 0.2 $$ per hour. If the initial population is $$ P_0$ $$, what is the population after 5 hours?

$$ P_0$ \times e$$^{-0.2 \times 5}$$ $$

Q: A patient receives a medication that decays in the bloodstream at a continuous rate of $$ r = 0.1 $$ per hour. If the initial dose is 200 mg, how much remains after 8 hours?

$$ 200 \times e$$^{-0.1 \times 8}$$ $$

Q: If the value of an investment grows continuously at a rate $$ r $$, which formula gives the value after $$ t $$ years?

$$ P = P_0$ \times e$$^{rt}$$ $$

Q: A city’s population is currently 500,000 and grows continuously at a rate of 2% per year. What will the population be after 15 years?

$$ 500,000 \times e$$^{0.02 \times 15}$$ $$

Question 1

If money is invested at a continuous compound interest rate of 5.5%, how long will it take for the investment to triple in value?

Accepted Answer

$$ t = \frac{\ln 3}{0.055} $$

Question 2

The continuous compound rate of decay of carbon-14 per year is $$ r = 0.0001238 . $$How long will it take for a sample to decay to half its original amount?

Accepted Answer

$$ t = \frac{\ln 2}{0.0001238} $$

Question 3

Suppose you invest $1000 at a continuous interest rate of 4% per year. What will be the value of your investment after 10 years?

Accepted Answer

$$ 1000 	imes e$$^{0.04 	imes 10}$$ $$

Question 4

A population of bacteria decays continuously at a rate of $$ r = 0.2 $$ per hour. If the initial population is $$ P_0$ $$, what is the population after 5 hours?

Accepted Answer

$$ P_0$ 	imes e$$^{-0.2 	imes 5}$$ $$

Question 5

If a radioactive substance decays continuously and its half-life is 5730 years, what is its decay rate $$ r $$?

Accepted Answer

$$ r = \frac{\ln 2}{5730} $$

Question 6

A patient receives a medication that decays in the bloodstream at a continuous rate of $$ r = 0.1 $$ per hour. If the initial dose is 200 mg, how much remains after 8 hours?

Accepted Answer

$$ 200 	imes e$$^{-0.1 	imes 8}$$ $$

Question 7

If the value of an investment grows continuously at a rate $$ r $$, which formula gives the value after $$ t $$ years?

Accepted Answer

$$ P = P_0$ 	imes e$$^{rt}$$ $$

Question 8

A scientist wants to know how long it will take for a sample to decrease to 25% of its original amount if it decays continuously at rate $$ r . $$What is the correct formula?

Accepted Answer

$$ t = \frac{\ln 0.25}{-r} $$

Question 9

If a population doubles every 7 years under continuous growth, what is the growth rate $$ r $$?

Accepted Answer

$$ r = \frac{\ln 2}{7} $$

Question 10

A chemical reaction decreases the concentration of a reactant continuously at a rate of $$ r = 0.05 $$ per minute. How long will it take for the concentration to decrease to one-tenth of its original value?

Accepted Answer

$$ t = \frac{\ln 0.1}{-0.05} $$

Question 11

A city’s population is currently 500,000 and grows continuously at a rate of 2% per year. What will the population be after 15 years?

Accepted Answer

$$ 500,000 	imes e$$^{0.02 	imes 15}$$ $$

Question 12

If a medication’s concentration halves every 3 hours, what is the continuous decay rate $$ r $$?

Accepted Answer

$$ r = \frac{\ln 2}{3} $$

Question 13

A scientist observes that a radioactive isotope decays to 60% of its original amount in 5 years. What is the continuous decay rate $$ r $$?

Accepted Answer

$$ r = \frac{\ln 0.6}{5} $$

Question 14

If a population grows continuously at rate $$ r $$, and after $$ t $$ years it has tripled, which equation relates $$ r $$ and $$ t $$?

Accepted Answer

$$ 3 = e$$^{rt}$$ $$

Question 15

A sample of a chemical decays continuously at rate $$ r . If $$after $$ t $$ years only $$ x $$ percent remains, which formula gives $$ t $$?

Accepted Answer

$$ t = \frac{\ln (x/100)}{-r} $$

Applications of Exponential Functions in Continuous Compounding and Radioactive Decay

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