Anything that grows at a rate proportional to its own size doubles in a fixed time, whatever it started at. That one fact is what half-life and doubling time both mean. Newton's law of cooling is the same idea applied to the gap between a cup and the room.
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Solve problems with a half-life, doubling time or continuous rate
A radioactive sample halves every 6 days. It starts at 640 grams. How many days until 20 grams are left?
A substance has a half-lifeof 25 years. How much of a 50-gram sample is left after 15 years? Round to two decimal places.
The activity of a source halves every 3 years. It starts at 160 becquerels. How many years until the activity is 10 becquerels?
An investment of $2,000 earns 4% interest, compounded continuously. How long until it grows to $6,000? Round to two decimal places.
A bacterial culture doubles every 2 hours. It starts at 320 bacteria. How many hours until there are 2,560 bacteria?
A substance decays continuously at 15% a year. What is its half-life? Round to two decimal places.
An investment doubles every 5 years. It starts at $80. How many years until it is worth $640?
A culture doubles every 4 hours. How long does it take to triple? Round to two decimal places.
A dose of a drug in the blood halves every 10 hours. It starts at 128 milligrams. How many hours until 8 milligrams are left?
A substance has a half-lifeof 30 years. How much of an 80-gram sample is left after 14 years? Round to two decimal places.
Use Newton's law of cooling
An 86∘C bowl of soup cools in a 22∘C room. After 12 minutes it is 57∘C. When is it 39∘C, to the nearest tenth of a minute?
A 92∘C cup of tea cools in a 17∘C room. After 8 minutes it is 51∘C.Find k inT(t)=17+75e−kt, to three decimal places.
An 88∘C mug of cocoa cools in a 20∘C room. After 9 minutes it is 70∘C. When is it 35∘C, to the nearest tenth of a minute?
A 91∘C cup of coffee cools in a 21∘C room. After 9 minutes it is 68∘C.Find k inT(t)=21+70e−kt, to three decimal places.
An 84∘C mug of cocoa cools in an 18∘C room. After 10 minutes it is 70∘C. When is it 25∘C, to the nearest tenth of a minute?
Model growth from two data points
In 2000, the population was 500; by 2004 it had grown exponentially to 875. Estimate the population in 2006, to the nearest whole number.
The rabbit colony was 800 in 2002 and 1,800 in 2004. Assuming exponential growth, what was it in 2000, to the nearest whole number?
Model the number of fish with P(t)=P0ekt. It was 400 in 2005 and 600 in 2015. Use the model to find it in 2010, to the nearest whole number.
Between 1995 and 2001, the town's population grew exponentially from 800 to 1,200. At that rate, what was the town's population in 1989, to the nearest whole number?
A herd of deer grows exponentially: 300 in 1995 and 675 in 1999. Find it in 1997.
Solve problems with a half-life, doubling time or continuous rate
A radioactive sample halves every 6 days. It starts at 640 grams. How many days until 20 grams are left?30 days
A substance has a half-lifeof 25 years. How much of a 50-gram sample is left after 15 years? Round to two decimal places.32.99 grams
The activity of a source halves every 3 years. It starts at 160 becquerels. How many years until the activity is 10 becquerels?12 years
An investment of $2,000 earns 4% interest, compounded continuously. How long until it grows to $6,000? Round to two decimal places.27.47 years
A bacterial culture doubles every 2 hours. It starts at 320 bacteria. How many hours until there are 2,560 bacteria?6 hours
A substance decays continuously at 15% a year. What is its half-life? Round to two decimal places.4.62 years
An investment doubles every 5 years. It starts at $80. How many years until it is worth $640?15 years
A culture doubles every 4 hours. How long does it take to triple? Round to two decimal places.6.34 hours
A dose of a drug in the blood halves every 10 hours. It starts at 128 milligrams. How many hours until 8 milligrams are left?40 hours
A substance has a half-lifeof 30 years. How much of an 80-gram sample is left after 14 years? Round to two decimal places.57.89 grams
Use Newton's law of cooling
An 86∘C bowl of soup cools in a 22∘C room. After 12 minutes it is 57∘C. When is it 39∘C, to the nearest tenth of a minute?26.4 minutes
A 92∘C cup of tea cools in a 17∘C room. After 8 minutes it is 51∘C.Find k inT(t)=17+75e−kt, to three decimal places.0.099
An 88∘C mug of cocoa cools in a 20∘C room. After 9 minutes it is 70∘C. When is it 35∘C, to the nearest tenth of a minute?44.2 minutes
A 91∘C cup of coffee cools in a 21∘C room. After 9 minutes it is 68∘C.Find k inT(t)=21+70e−kt, to three decimal places.0.044
An 84∘C mug of cocoa cools in an 18∘C room. After 10 minutes it is 70∘C. When is it 25∘C, to the nearest tenth of a minute?94.1 minutes
Model growth from two data points
In 2000, the population was 500; by 2004 it had grown exponentially to 875. Estimate the population in 2006, to the nearest whole number.about 1,158
The rabbit colony was 800 in 2002 and 1,800 in 2004. Assuming exponential growth, what was it in 2000, to the nearest whole number?about 356
Model the number of fish with P(t)=P0ekt. It was 400 in 2005 and 600 in 2015. Use the model to find it in 2010, to the nearest whole number.about 490
Between 1995 and 2001, the town's population grew exponentially from 800 to 1,200. At that rate, what was the town's population in 1989, to the nearest whole number?about 533
A herd of deer grows exponentially: 300 in 1995 and 675 in 1999. Find it in 1997.450
A culture doubles every 3 hours. It starts at 80 bacteria. How many after 12 hours?
Twelve hours is four doubling periods.
So the count is 80 times 2 to the fourth.
That is 1280.
The answer is 1,280 bacteria.
Where students go wrong
Multiplying by the number of periods instead of raising to it. Four doublings is not four times as many, it is sixteen times as many, because each doubling acts on what the last one left.
Also called half life, doubling time, continuous growth, radioactive decay or population growth.
What you can put on this worksheet
Exponential growth and decay
A dose of a drug in the blood halves every 8 hours. It starts at 192 milligrams. How many milligrams are left after 16 hours?→ 48 milligrams
Newton's law of cooling
An 81∘C mug of cocoa cools in a 23∘C room. By Newton's law of cooling its temperature is T(t)=23+58e−0.05tafter t minutes. What is it after 14 minutes, to the nearest tenth of a degree?→ 51.8∘C
From two data points
The herd of deer was 600 in 2005 and 720 in 2008. Assuming exponential growth, what will it be in 2014, to the nearest whole number?→ about 1,037
Continuous rates, compounding, half-lives, doubling to tripling
A 200-gram sample decays continuously at 10% a day. How many grams are left after 2 days? Round to two decimal places.→ 163.75 grams
Questions about these worksheets
Yes. Every half-life and doubling time sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the half-life and doubling time questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 4: exponential growth and decay, newton's law of cooling, from two data points and continuous rates, compounding, half-lives, doubling to tripling. Tick as many as you want and set how many of each, or let it spread them evenly.
Multiplying by the number of periods instead of raising to it. Four doublings is not four times as many, it is sixteen times as many, because each doubling acts on what the last one left.