Exponential Growth and Decay worksheet with four math problems involving population, bacteria, radioactive decay, and compound interest.

We are given 4 exponential growth/decay problems. All involve the general formula for continuous exponential change:

> P(t) = P₀ · e^(kt)

Where:
- P(t) is the amount at time *t*
- P₀ is the initial amount
- k is the growth/decay rate (positive for growth, negative for decay)
- t is time (in the same units as the rate)

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Problem 1: Island Population Growth

> Growth rate = 3.8% per year → k = 0.038
> Initial population P₀ = 1543
> Time t = 5.2 years

Formula:
P(5.2) = 1543 · e^(0.038 × 5.2)

Calculate exponent:
0.038 × 5.2 = 0.1976

Compute:
e^0.1976 ≈ 1.2185 (using calculator)

Final value:
1543 × 1.2185 ≈ 1880.15

✔ Answer: Approximately 1880 people

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Problem 2: E. coli Bacteria Growth

> Growth rate = 1.9% per minute → k = 0.019
> Initial population P₀ = 172.0 million
> Time t = 7.2 minutes

Formula:
P(7.2) = 172.0 · e^(0.019 × 7.2)

Calculate exponent:
0.019 × 7.2 = 0.1368

Compute:
e^0.1368 ≈ 1.1465

Final value:
172.0 × 1.1465 ≈ 197.198 million

✔ Answer: Approximately 197.2 million

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Problem 3: Carbon-14 Decay

> Decay rate = 12.10% per thousand years → Since it’s decay, k = -0.1210
> Initial mass P₀ = 135.2 mg
> Time t = 2.2 thousand years

Formula:
P(2.2) = 135.2 · e^(-0.1210 × 2.2)

Calculate exponent:
-0.1210 × 2.2 = -0.2662

Compute:
e^(-0.2662) ≈ 0.7662

Final value:
135.2 × 0.7662 ≈ 103.6 mg

✔ Answer: Approximately 103.6 mg

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Problem 4: Savings Account (Continuous Compounding)

> Interest rate = 3.1% per year → k = 0.031
> Initial balance P₀ = $1077.00
> Final balance P(t) = $1486.73
> Find time t

Formula:
1486.73 = 1077.00 · e^(0.031·t)

Divide both sides by 1077.00:
1486.73 / 1077.00 ≈ 1.37997

So:
e^(0.031·t) = 1.37997

Take natural log of both sides:
ln(1.37997) = 0.031·t

Compute ln(1.37997):
≈ 0.3215

Then:
t = 0.3215 / 0.031 ≈ 10.37 years

✔ Answer: Approximately 10.37 years

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## ✔ Final Answers:

1. 1880 people
2. 197.2 million bacteria
3. 103.6 mg
4. 10.37 years

Let me know if you’d like to see step-by-step calculations with more decimal precision or graphs!