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

NameDate

Figure — use these to answer the problems

Make the exponent negative and growth becomes decay. The same equation covers both.

y = 1 · 0.5^x + 0

the starting amount1

k, the rate constant0.5

the time shown1

Figure 1

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
2
Why is Newton's law of cooling the same equation as decay?
- It is the temperature gap, not the temperature, that decays exponentially.
- It is a coincidence of the formulas.
100 grams with a half-life of 3 hours. How much remains after 6 hours?
Answer
64 grams with a half-life of 2 years. How much remains after 8 years?
Answer
dy/dt = ky with k = −0.3. Is this growth or decay? Enter 1 for growth, 2 for decay.
Answer

The same ideas with more to keep track of.

3 problems

dT/dt = −k(T − 18). What is the ambient temperature?
Answer
A population doubles every 10 years. How many times larger is it after 30 years?
Answer
An object at exactly the ambient temperature. What is dT/dt?
Answer

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

4 problems

9
Match each situation to the sign of its rate constant.
Draw a line from each item on the left to its match on the right.
- A growing bacterial colony
- A decaying radioactive sample
- An object already at room temperature
- Positive
- Negative
- The rate itself is zero
32 grams with a half-life of 1 year. How much remains after 5 years?
Answer
The Sample
200 grams with a half-life of 4 days. How many grams remain after 12 days?
Answer
The Coffee
Coffee cooling in a room at 21 degrees. What temperature does it approach in the long run?
Answer