Exponential Growth Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Exponential Growth.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Exponential growth occurs when a quantity increases by a constant multiplicative factor over equal intervals.

Exponential growth means the amount added each period is proportional to the current amount — the bigger it gets, the faster it grows, creating an accelerating curve.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: f(t) = A \cdot b^t with b > 1: the growth rate at any moment is proportional to the current value f(t), giving f'(t) = k \cdot f(t) for some constant k > 0.

Common stuck point: Students model exponential situations with linear equations.

Sense of Study hint: Look for constant percent change; if yes, use a base multiplier model.

Worked Examples

Example 1

easy
A population of bacteria doubles every 3 hours. If there are initially 500 bacteria, how many will there be after 12 hours?

Solution

  1. 1
    The exponential growth model is P(t) = P_0 \cdot 2^{t/d}, where d is the doubling time.
  2. 2
    Substitute: P(12) = 500 \cdot 2^{12/3} = 500 \cdot 2^4.
  3. 3
    P(12) = 500 \cdot 16 = 8{,}000.

Answer

8{,}000 \text{ bacteria}
Exponential growth occurs when a quantity increases by a constant percentage over equal time intervals. The doubling-time formula P(t) = P_0 \cdot 2^{t/d} is a special case of P(t) = P_0 \cdot b^t. In 12 hours, the population doubles 4 times: 500 \to 1000 \to 2000 \to 4000 \to 8000.

Example 2

medium
A city's population grows at 3\% per year. If the current population is 200{,}000, when will it reach 300{,}000?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
An investment grows from \1{,}000 to \1{,}500 in 5 years with continuous compounding. Find the annual growth rate.

Example 2

hard
Two populations grow exponentially: P_1(t) = 100 \cdot 3^t and P_2(t) = 5000 \cdot (1.2)^t. When will P_1 overtake P_2?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

exponential functiongrowth vs decaycompound interest