dy/dt = ky says the rate of change is proportional to the amount present. Its solution is y = y₀e^(kt).
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The sign of k
A positive k gives growth and a negative one gives decay. Nothing else about the equation changes.
Half-life
The time to halve is fixed and does not depend on how much you started with. That is the signature of exponential decay.
Newton's law of cooling
dT/dt = −k(T − A). The rate depends on the gap between the object and the ambient temperature A.
Why it is the same equation
Substituting u = T − A turns cooling into du/dt = −ku. It is exponential decay of the temperature gap.
Where it stops
The object approaches the ambient temperature but never quite reaches it. A is the stable equilibrium.
One equation behind three phenomena
dy/dt = ky says the rate is proportional to the amount. It produces exponential growth for positive k and decay for negative, and it is the same equation in both cases.
Half-life is independent of the starting amount
The time to halve depends only on k, not on how much you began with. That is a distinctive signature of exponential decay and is what makes radiometric dating possible.
Newton's law of cooling
An object cools at a rate proportional to the difference from ambient temperature. The equation is dT/dt = k(T − Tₐ), and its solutions approach ambient asymptotically without ever arriving.
Where proportionality fails
Populations run out of resources, so unbounded exponential growth is always a short-term approximation. Recognising the limits of the model is part of using it responsibly.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 0.50)
Step 3: Watch an Example
One step at a time.
Watch Amara Use a Half-Life
Amara has 80 grams of a substance with a half-life of 5 years and wants the amount after 15 years.
- Step 1
She divides 15 by 5, finding that three half-lives have passed.
Step 4: Your Turn
Practice makes it stick.
The Sample
Problem 1 of 2
200 grams with a half-life of 4 days. How many grams remain after 12 days?
The Coffee
Problem 2 of 2
Coffee cooling in a room at 21 degrees. What temperature does it approach in the long run?
Model It
1 of 8
100 grams with a half-life of 3 hours. How much remains after 6 hours?
2 of 8
64 grams with a half-life of 2 years. How much remains after 8 years?
3 of 8
dy/dt = ky with k = −0.3. Is this growth or decay? Enter 1 for growth, 2 for decay.
4 of 8
dT/dt = −k(T − 18). What is the ambient temperature?
5 of 8
A population doubles every 10 years. How many times larger is it after 30 years?
6 of 8
An object at exactly the ambient temperature. What is dT/dt?
7 of 8
Match each situation to the sign of its rate constant.
Tap a card on the left to start.
8 of 8
32 grams with a half-life of 1 year. How much remains after 5 years?
Step 5: Quick Check
Show what you know.
Question 1 of 2
160 grams with a half-life of 5 days. How much remains after 15 days?
Question 2 of 2
Why is Newton's law of cooling the same equation as decay?
What You Learned
- dy/dt = ky solves to y = y₀e^(kt): positive k grows, negative k decays.
- A half-life is fixed regardless of the starting amount.
- Newton's law of cooling is exponential decay of the gap to the ambient temperature.