1. Negative Exponent: For decay, your $r$ must be negative in the formula ($A = Pe^{-rt}$).
2. The Half-Life Rule: To find $r$ from a half-life, use $r = 0.693 / T_{1/2}$.
3. Check the Remainder: Your final answer must always be smaller than your starting amount.
4. The Asymptote: Remember that decay never hits zero. If your calculator says "0", it's just rounding!
Calculate the remaining amount ($A$) using $A = Pe^{-rt}$.
The Medicine: You take 400mg of a painkiller. It decays in your body at a rate of 25% per hour ($r=0.25$). How much is left after 3 hours?
The Fading Signal: A radio signal has a power of 1,000 watts. As it travels through a storm, it decays at a rate of 0.1% per kilometer ($r=0.001$). What is the power after 500 kilometers?
Find the "Half-Life Clock" for these substances.
The Isotope: A radioactive element decays at a rate of 12% per day ($r=0.12$). What is its half-life?
The Memory: A student forgets 5% of a lesson every day if they don't review it. What is the "Half-Life" of the lesson in their mind?
If a substance has a half-life of 10 years, and you start with 1,000 grams... will you have 0 grams after 20 years? Explain why or why not.
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The Ancient Scroll: An archaeologist finds a scroll that has 70% of its original Carbon-14 left. The decay rate ($r$) for Carbon-14 is $0.000121$ per year.
How old is the scroll?
(Hint: Solve $0.70 = 1.00 \cdot e^{-0.000121t}$ using your $\ln$ button if you know how, or guess and check!)
A cup of cocoa is $180^\circ$F. The room is $70^\circ$F. The cocoa cools at a rate of 10% per minute ($r=0.1$).
The formula is: $Temp = 70 + (180 - 70)e^{-0.1t}$.
Task: Find the temperature after 10 minutes. Will it ever reach $60^\circ$F? Why?
Objective: Perform the M&M (or coin) decay simulation with a younger sibling.
The Activity:
1. Start with 40 coins/candies.
2. Shake them. Remove all "Tails."
3. Count what's left. Repeat until only 1 or 2 are left.
4. Plot the "Population" after each shake on a graph.
The Lesson: "Even when things go away, they go away in a pattern that we can predict. God's math is even in the 'disappearing'."
Did the younger student enjoy the 'decay' (eating the candies)? [ ] Yes [ ] No
What was their observation about the graph? _________________________