Section 6.4: Applications & Word Problems

This section applies exponential functions to real-world problems, including population growth, radioactive decay, and compound interest.

Example 1: Population Growth

The population of a town grows according to $ P(t) = 5000 \cdot 1.03^t $, where t is years. Find the population after 10 years.

Step 1: Substitute t = 10 → $ P(10) = 5000 \cdot 1.03^{10} $

Step 2: Calculate → $ P(10) \approx 5000 \cdot 1.3439 \approx 6719.5 $

Step 3: Population ≈ 6720

A principal of $2000 is invested at 5% annual interest, compounded quarterly. Find the amount after 5 years.

Step 1: Use formula $ A = P(1 + r/n)^{nt} $

Step 2: $ A = 2000(1 + 0.05/4)^{4\cdot5} = 2000(1.0125)^{20} $

Step 3: $ A \approx 2000 \cdot 1.2820 \approx 2564 $

Radioactive decay: $ N(t) = 100 \cdot (1/2)^{t/5} $. Find N after 10 years.
Population doubles every 8 years. If current population is 12,000, find population after 16 years.
Invest $1500 at 6% interest, compounded monthly. Find amount after 3 years.
A bacteria culture grows according to $ B(t) = 50 \cdot 2^t $. Find B after 6 hours.
A city's population decreases by 2% yearly. Find population after 5 years if current population is 50,000.