Section 6.5: Applications of Exponentials

Exponential functions are widely used to model population growth, radioactive decay, compound interest, and other real-life situations.

Example 1

A population of 1,000 bacteria doubles every 3 hours. Find the population after 9 hours.

Population formula: \( P(t) = P_0 \cdot 2^{t/3} \)

Substitute \( t = 9 \): \( P(9) = 1000 \cdot 2^{9/3} = 1000 \cdot 2^3 = 1000 \cdot 8 = 8000 \)

Example 2

A $5,000 investment earns 4% annual interest compounded yearly. Find the amount after 5 years.

Compound interest formula: \( A = P(1 + r)^t \)

Substitute values: \( A = 5000 \cdot (1 + 0.04)^5 \approx 5000 \cdot 1.2167 \approx 6083.50 \)

Practice Problems

  1. A population of 500 rabbits triples every 4 months. Find the population after 1 year.
  2. A $2,000 investment earns 5% interest compounded annually. Find the amount after 10 years.
  3. Radioactive substance has half-life of 6 hours. If you start with 80g, how much remains after 18 hours?
  4. A town’s population grows at 3% annually. Starting at 10,000, what is the population after 7 years?
  5. A bacteria culture increases by 150% every hour. Find the population after 3 hours if it starts at 200.
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