Bacterial Colony Growth

A bacteria colony starts with 500 cells and doubles every hour. How many cells will there be after 6 hours?

Identify the values: $A_0 = 500$ (initial), $b = 2$ (doubles), $t = 6$ hours = Values identified

Write the formula: $A(t) = A_0 \cdot b^t = 500 \cdot 2^6$ = Formula set up

Calculate $2^6$: $2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$ = $2^6 = 64$

Multiply: $A(6) = 500 \cdot 64 = 32000$ = 32000 cells

Answer: After 6 hours, the colony will have **32,000 cells**.

Investment with Compound Growth

You invest 1000 euros at 8% annual interest, compounded yearly. How much will you have after 10 years?

Identify the values: $A_0 = 1000$ euros, $r = 0.08$ (8%), $t = 10$ years = Values identified

Find the growth factor: $b = 1 + r = 1 + 0.08 = 1.08$ = $b = 1.08$

Apply the formula: $A(10) = 1000 \cdot (1.08)^{10}$ = Formula applied

Calculate $(1.08)^{10}$: $(1.08)^{10} \approx 2.159$ = Growth multiplier found

Find the final amount: $A(10) = 1000 \cdot 2.159 \approx 2159$ euros = 2159 euros

Answer: After 10 years, the investment grows to approximately **2159 euros**, more than doubling!

Finding the Growth Rate

A town's population grew from 10,000 to 16,000 in 5 years. What is the annual growth rate?

Set up the equation: $16000 = 10000 \cdot b^5$ = Equation formed

Isolate $b^5$: $b^5 = \frac{16000}{10000} = 1.6$ = $b^5 = 1.6$

Solve for $b$: $b = \sqrt[5]{1.6} = 1.6^{0.2} \approx 1.0986$ = $b \approx 1.0986$

Find the rate: $r = b - 1 = 1.0986 - 1 = 0.0986 \approx 9.86\%$ = About 9.86% per year

Answer: The town grew at approximately **9.86% per year**.

Mistake: Confusing growth factor with growth rate

Why: If something grows by 5%, the growth rate is $r = 0.05$, but the growth factor is $b = 1.05$. Students often use 0.05 as the base instead of 1.05.

Correct: Growth factor = $1 +$ growth rate. If the rate is 5%, use $b = 1.05$ in the formula.

Mistake: Thinking exponential growth is always fast

Why: Early exponential growth can look slow. $2^1 = 2$, $2^2 = 4$, $2^3 = 8$ seems modest, but it accelerates dramatically.

Correct: Exponential growth starts slowly but accelerates. The magic is in the later stages: $2^{20} > 1,000,000$!

Mistake: Using addition instead of multiplication

Why: Linear thinking is intuitive: 'Add 10% each year' but exponential growth multiplies: $100 \to 110 \to 121 \to 133.1$

Correct: Each period, multiply by the growth factor: $A_{new} = A_{old} \cdot b$

Compound Interest and Retirement

Banks use exponential growth for compound interest. Starting early makes a huge difference due to the 'snowball effect'.

Investing 5000 euros at 7% annual return: after 30 years you'll have $5000 \cdot 1.07^{30} \approx 38,061$ euros!

Population Growth Models

Biologists model population growth under ideal conditions (unlimited resources) as exponential.

If a rabbit population of 100 grows at 50% per year: after 4 years there would be $100 \cdot 1.5^4 = 506$ rabbits.

Technology and Moore's Law

Gordon Moore observed that computing power doubles roughly every 2 years, an exponential pattern that held for decades.

If a computer has 1 million transistors today and doubles every 2 years, in 20 years it will have $1,000,000 \cdot 2^{10} \approx 1$ billion transistors.

Q: What's the difference between exponential and linear growth?

A: Linear growth adds a constant amount each period (like $+10$ per year). Exponential growth multiplies by a constant factor (like $\times 1.1$ per year). Over time, exponential growth far outpaces linear growth.

Q: What does 'doubling time' mean?

A: Doubling time is how long it takes for a quantity to double. For 7% annual growth, the doubling time is approximately $\frac{70}{7} = 10$ years (Rule of 70).

Q: Can exponential growth continue forever?

A: In the real world, no. Resources are limited, so exponential growth eventually slows (called logistic growth). But the exponential model works well for early growth phases.