Growth, Decay and Cooling

One equation behind three familiar phenomena.

12 problems · about 23 minutes · F-LE.A.1, F-LE.A.4, A-CED.A.1

What this lesson teaches

The student models exponential growth, decay and Newton cooling with differential equations.

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.

Straightforward practice. Get the method working first.

5 problems

160 grams with a half-life of 5 days. How much remains after 15 days?
Answer 20
Why 20 grams.
Why is Newton's law of cooling the same equation as decay?
Answer It is the temperature gap, not the temperature, that decays exponentially.
Why The gap decays exponentially.
100 grams with a half-life of 3 hours. How much remains after 6 hours?
Answer 25
Why Two half-lives.
64 grams with a half-life of 2 years. How much remains after 8 years?
Answer 4
Why Four half-lives.
dy/dt = ky with k = −0.3. Is this growth or decay? Enter 1 for growth, 2 for decay.
Answer 2
Why A negative rate constant.

The same ideas with more to keep track of.

3 problems

dT/dt = −k(T − 18). What is the ambient temperature?
Answer 18
Why Read it from the bracket.
A population doubles every 10 years. How many times larger is it after 30 years?
Answer 8
Why Three doublings.
An object at exactly the ambient temperature. What is dT/dt?
Answer 0
Why The gap is zero.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each situation to the sign of its rate constant.
Answer A growing bacterial colony → Positive; A decaying radioactive sample → Negative; An object already at room temperature → The rate itself is zero
Why One of the three is not changing at all.
32 grams with a half-life of 1 year. How much remains after 5 years?
Answer 1
Why Five halvings.
The Sample: 200 grams with a half-life of 4 days. How many grams remain after 12 days?
Answer 25
Why 25 grams.
The Coffee: Coffee cooling in a room at 21 degrees. What temperature does it approach in the long run?
Answer 21
Why 21 degrees.