1. Locate $e$: $e$ is a constant ($\approx 2.718$). Do not solve for it.
2. Identify Continuous: If a problem says "Continuous," use $A = Pe^{rt}$.
3. Exponent First: Multiply the rate ($r$) by the time ($t$) before pressing the $e^x$ button.
4. Check for Abundance: Continuous growth should always be larger than discrete growth for the same rate.
Calculate the final amount ($A$) using the formula $A = Pe^{rt}$.
The Abundant Savings: You invest $\$5,000$ at a rate of 5% ($r=0.05$) compounded continuously for 12 years. How much do you have?
The City of Light: A city has 100,000 people. It grows continuously at a rate of 3% per year. What will the population be after 25 years?
The Comparison: You have $\$1,000$ at 8% interest for 5 years.
A) Calculate $A$ if compounded Monthly ($n=12$).
B) Calculate $A$ if compounded Continuously ($e$).
How much "Abundance" ($B - A$) did the continuous path provide?
If you have a 100% interest rate and $\$1$... why don't you have infinite money if it compounds infinitely often? What is the limit of that $\$1$?
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The Bacteria Bloom: A culture of 500 bacteria grows continuously at a rate of 40% per hour. How many bacteria are present after 6 hours? After 24 hours?
The Rising Dough: A baker's dough expands at a rate of 15% per hour. If the dough starts at a volume of 2 liters, what is the volume after 3 hours of rising?
If something grows continuously at a rate of $r$, the time it takes to double ($A = 2P$) can be found using the formula: $t = \frac{\ln(2)}{r}$.
(Note: $\ln(2) \approx 0.693$)
Task: If the Gospel spreads in a region at a rate of 10% per year ($r=0.10$), how many years will it take for the number of believers to double?
Compare this to a rate of 20%. Does doubling the rate cut the time in half?
Objective: Explain "Continuous" vs "Steps" to a younger sibling.
Activity: Use a faucet.
1. Fill a cup by pouring a little bit, stopping, waiting, then pouring again (Steps).
2. Fill a cup by letting the water flow in a steady stream (Continuous).
The Lesson: "God's love is like the steady stream. It doesn't wait for us to be good or for a certain day of the week. It flows every second, and that's why it's the most powerful way to grow."
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