Q: A disease spreads through a community. On March 1, there were 100 infected people. The number of infected people doubles every 3 days. How many infected people will there be after 12 days?

$$100 \times 2$$^{12/3}$$ = 100 \times 2^4$ = 1600$$

Q: If a radioactive isotope has a half-life of 10 years, how much of the isotope remains after 20 years if you start with 50 grams?

$$50 \times \left(\frac{1}{2}\$$right)^{20/10}$$ = 50 \times \left(\frac{1}{2}\$$right)^2$$ = 12.5$$ grams

Q: A substance has a half-life of 7 years. If you start with 80 grams, how much remains after 14 years?

$$80 \times \left(\frac{1}{2}\$$right)^{14/7}$$ = 80 \times \left(\frac{1}{2}\$$right)^2$$ = 20$$ grams

Q: If there are initially 70 grams of a substance with a half-life of 5 years, how much remains after $$t$$ years?

$$70 \times \left(\frac{1}{2}\$$right)^{t/5}$$

Q: A bone is found with 2% of the usual amount of carbon-14. The half-life of carbon-14 is 5730 years. How old is the bone?

$$t = 5730 \times \log_2$\left(\frac{1}{0.02}\right) \approx 5730 \times 5.6439 \approx 32,370$$ years

Q: A scientist starts with 100 grams of a radioactive substance with a half-life of 8 years. How many years will it take for only 25 grams to remain?

$$t = 8 \times \log_2$\left(\frac{100}{25}\right) = 8 \times 2 = 16$$ years

Q: A population of bacteria doubles every 6 hours. If there are initially 500 bacteria, how many will there be after 18 hours?

$$500 \times 2$$^{18/6}$$ = 500 \times 2^3$ = 4000$$

Q: A chemical reaction produces a compound that decays with a half-life of 4 hours. If you start with 40 grams, how much remains after 12 hours?

$$40 \times \left(\frac{1}{2}\$$right)^{12/4}$$ = 40 \times \left(\frac{1}{2}\$$right)^3$$ = 5$$ grams

Q: A city has a population of 20,000 people. If the population doubles every 10 years, what will the population be after 30 years?

$$20,000 \times 2$$^{30/10}$$ = 20,000 \times 2^3$ = 160,000$$

Question 1

A disease spreads through a community. On March 1, there were 100 infected people. The number of infected people doubles every 3 days. How many infected people will there be after 12 days?

Accepted Answer

$$100 	imes 2$$^{12/3}$$ = 100 	imes 2^4$ = 1600$$

Question 2

A colony of mold is growing on a cheeseburger. When discovered, it has a mass of 10 mg. One week later, it has a mass of 30 mg. What is the doubling time, measured in days?

Accepted Answer

$$K = \frac{7}{\log_2$(30/10)} = \frac{7}{\log_2$(3)} \approx 4.42$$ days

Question 3

If a radioactive isotope has a half-life of 10 years, how much of the isotope remains after 20 years if you start with 50 grams?

Accepted Answer

$$50 	imes \left(\frac{1}{2}\$$right)^{20/10}$$ = 50 	imes \left(\frac{1}{2}\$$right)^2$$ = 12.5$$ grams

Question 4

A substance has a half-life of 7 years. If you start with 80 grams, how much remains after 14 years?

Accepted Answer

$$80 	imes \left(\frac{1}{2}\$$right)^{14/7}$$ = 80 	imes \left(\frac{1}{2}\$$right)^2$$ = 20$$ grams

Question 5

If there are initially 70 grams of a substance with a half-life of 5 years, how much remains after $$t$$ years?

Accepted Answer

$$70 	imes \left(\frac{1}{2}\$$right)^{t/5}$$

Question 6

A bone is found with 2% of the usual amount of carbon-14. The half-life of carbon-14 is 5730 years. How old is the bone?

Accepted Answer

$$t = 5730 	imes \log_2$\left(\frac{1}{0.02}ight) \approx 5730 	imes 5.6439 \approx 32,370$$ years

Question 7

A scientist starts with 100 grams of a radioactive substance with a half-life of 8 years. How many years will it take for only 25 grams to remain?

Accepted Answer

$$t = 8 	imes \log_2$\left(\frac{100}{25}ight) = 8 	imes 2 = 16$$ years

Question 8

A population of bacteria doubles every 6 hours. If there are initially 500 bacteria, how many will there be after 18 hours?

Accepted Answer

$$500 	imes 2$$^{18/6}$$ = 500 	imes 2^3$ = 4000$$

Question 9

A chemical reaction produces a compound that decays with a half-life of 4 hours. If you start with 40 grams, how much remains after 12 hours?

Accepted Answer

$$40 	imes \left(\frac{1}{2}\$$right)^{12/4}$$ = 40 	imes \left(\frac{1}{2}\$$right)^3$$ = 5$$ grams

Question 10

A city has a population of 20,000 people. If the population doubles every 10 years, what will the population be after 30 years?

Accepted Answer

$$20,000 	imes 2$$^{30/10}$$ = 20,000 	imes 2^3$ = 160,000$$

Question 11

A scientist observes a sample of radioactive material. After 15 years, only 10 grams remain from an initial 80 grams. What is the half-life of the material?

Accepted Answer

$$K = \frac{15}{\log_2$(80/10)} = \frac{15}{\log_2$(8)} = \frac{15}{3} = 5$$ years

Question 12

A medication loses half its effectiveness every 2 days. If a patient starts with a dose of 100 mg, how much will remain effective after 6 days?

Accepted Answer

$$100 	imes \left(\frac{1}{2}\$$right)^{6/2}$$ = 100 	imes \left(\frac{1}{2}\$$right)^3$$ = 12.5 mg$$

Question 13

A radioactive sample decays to 25% of its original mass in 12 years. What is the half-life of the sample?

Accepted Answer

$$K = \frac{12}{\log_2$(1/0.25)} = \frac{12}{2} = 6$$ years

Question 14

A patient receives a radioactive tracer with a half-life of 3 hours. If the initial dose is 60 mg, how much remains after 9 hours?

Accepted Answer

$$60 	imes \left(\frac{1}{2}\$$right)^{9/3}$$ = 60 	imes \left(\frac{1}{2}\$$right)^3$$ = 7.5 mg$$

Question 15

A population of rabbits triples every 5 years. If there are initially 200 rabbits, how many will there be after 15 years?

Accepted Answer

$$200 	imes 3$$^{15/5}$$ = 200 	imes 3^3$ = 5400$$

Question 16

A scientist wants to know how long it will take for a sample to decay to 10% of its original mass. The half-life is 6 years. What is the required time?

Accepted Answer

$$t = 6 	imes \log_2$\left(\frac{1}{0.1}ight) \approx 6 	imes 3.3219 \approx 19.93$$ years

Question 17

A sample of bacteria grows exponentially. If the population doubles every 2 hours and starts at 1000, how many bacteria will there be after 10 hours?

Accepted Answer

$$1000 	imes 2$$^{10/2}$$ = 1000 	imes 2^5$ = 32,000$$

Question 18

A radioactive isotope decays to 5 grams from an initial 40 grams in 16 years. What is the half-life?

Accepted Answer

$$K = \frac{16}{\log_2$(40/5)} = \frac{16}{3} \approx 5.33$$ years

Lecture 13 - Calculus Study Guide: Doubling Time and Half-Life Problems

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