1. Law of Proportionality: $\frac{dy}{dt} = ky$.
2. The Solution: $y = Ae^{kt}$ (where $A$ is the initial amount).
3. Find k: Use the given rate or a data point to solve for the constant.
4. Half-Life Rule: $k = \ln(0.5) / T_{1/2} \approx -0.693 / T_{1/2}$.
Using the equation $\frac{dy}{dt} = ky$ and $y(0) = A$.
The Seeds: A population of seeds grows at a rate proportional to its size. Initially, there are 100 seeds. After 2 hours, there are 400 seeds.
1. Solve for $k$.
2. Find the population after 5 hours.
The Doubling Life: A bacteria colony doubles every 3 hours ($T_{1/2} = 3$ for growth). Find the growth constant $k$.
Carbon Dating: Carbon-14 has a half-life of 5,730 years. Find the decay constant $k$.
The Sample: If an ancient scroll has 70% of its original Carbon-14 left... how old is it?
In the growth formula $Ae^{kt}$, what happens to the population if $k=0$? Does the system grow, decay, or stay the same? Why is this called the "Stagnant" solution?
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A cup of soup at $180^\circ$ is left in a $70^\circ$ room. After 10 minutes, the soup is $120^\circ$.
Formula: $T(t) = T_{\text{room}} + (T_0 - T_{\text{room}})e^{-kt}$.
1. Solve for $k$.
2. When will the soup be $80^\circ$?
A bank offers 10% interest.
Account A: Compounded Monthly ($n=12$).
Account B: Compounded Continuously ($\frac{dA}{dt} = 0.10A$).
Task: If you start with $\$1,000$, find the difference in the balances after 10 years.
Objective: Explain Proportional Growth to a younger sibling using a snowball.
The Activity:
1. Show them a tiny snowball. "If I roll this, it only picks up a few snowflakes."
2. Show them a big snowball. "If I roll this, it picks up a hundred snowflakes in one turn!"
The Lesson: "The more you have, the faster you grow. That's the Rule of the Snowball, and it's the Rule of God's Blessing!"
Response: ___________________________________________________________