Section 6.1: Growth & Decay Models

This section introduces exponential growth and decay models, including the general formulas and their applications in real-world contexts.

Example 1: Exponential Growth

A population of bacteria doubles every 3 hours. If there are initially 500 bacteria, how many will there be after 9 hours?

Step 1: Growth formula: $ P(t) = P_0 \cdot 2^{t/3} $

Step 2: Substitute values: $ P(9) = 500 \cdot 2^{9/3} = 500 \cdot 2^3 $

Step 3: Compute: $ 500 \cdot 8 = 4000 $

A radioactive substance has a half-life of 5 years. If the initial mass is 200 g, find the remaining mass after 15 years.

Step 1: Decay formula: $ M(t) = M_0 \cdot (1/2)^{t/5} $

Step 2: Substitute values: $ M(15) = 200 \cdot (1/2)^{15/5} = 200 \cdot (1/2)^3 $

Step 3: Compute: $ 200 \cdot 1/8 = 25 $ g

A town’s population grows by 5% per year. If the current population is 10,000, find the population after 6 years.
An investment of $1000 earns 4% interest compounded annually. Find the balance after 10 years.
A substance decays at 3% per year. If the initial amount is 500 g, find the remaining mass after 8 years.
The number of users on a website triples every 2 months. If the initial number is 200, find the number after 6 months.
A bacteria culture decreases by 20% each hour. If there are 1000 bacteria initially, how many remain after 5 hours?