Exponential Growth & Decay Word Problems

The exponential growth model

When a quantity doubles at a fixed interval, the formula is:

$N(t) = N_0 \cdot 2^{t/d}$

where $N_0$ is the initial amount, $d$ is the doubling time, and $t$ is the elapsed time (same units as $d$).

Step-by-step solution

Given: Initial population $N_0 = 500$. Doubling time $d = 3$ hours. Find $N(12)$.

Step 1 — Identify the formula: $N(t) = 500 \cdot 2^{t/3}$

Step 2 — Plug in $t = 12$ hours:

$N(12) = 500 \cdot 2^{12/3} = 500 \cdot 2^4$

Step 3 — Compute $2^4$: $2^4 = 16$.

Step 4 — Final answer: $N(12) = 500 \times 16 = 8{,}000$ bacteria.

Check by tracing doublings:

• $t = 0$: $500$
• $t = 3$: $1{,}000$ (first doubling)
• $t = 6$: $2{,}000$ (second doubling)
• $t = 9$: $4{,}000$ (third doubling)
• $t = 12$: $8{,}000$ (fourth doubling) ✓

The general exponential model

More generally: $N(t) = N_0 \cdot b^{t}$, where $b > 1$ is growth and $0 < b < 1$ is decay.

• Half-life version: $N(t) = N_0 \cdot (1/2)^{t/h}$ where $h$ = half-life.
• Percentage growth: If population grows 5% per year, $b = 1.05$, so $N(t) = N_0 \cdot 1.05^t$.

Common exam question types

• "How long until the population reaches 10,000?" → Set $N(t) = 10{,}000$, solve for $t$ using logarithms.
• "What is the growth rate?" → If $N(5) = 2N_0$, then doubling time is 5.

Try it in the visualization

Adjust the initial population and doubling time. The growth curve shows exponential increase. Each doubling is marked on the curve. Toggle "decay mode" to see half-life behavior instead.