Section 6.8: Real-Life Applications
Explore how exponential functions model real-world phenomena like population growth, compound interest, and radioactive decay.
Example 1: Population Growth
A town has 5,000 residents and its population doubles every 20 years. Write an exponential function for the population and find it after 60 years.
Exponential function: \( P(t) = 5000 \cdot 2^{t/20} \)
After 60 years: \( P(60) = 5000 \cdot 2^{60/20} = 5000 \cdot 2^3 = 5000 \cdot 8 = 40,000 \)
Example 2: Compound Interest
Invest $2,000 at 5% annually compounded yearly. Find the amount after 10 years.
Exponential function: \( A = 2000 (1 + 0.05)^{10} = 2000 \cdot 1.05^{10} \approx 3257.79 \)
Practice Problems
- Population of 800 bacteria doubles every 3 hours. Find the population after 9 hours.
- Invest $1,000 at 4% annually for 6 years, compounded yearly. Find total amount.
- A radioactive substance has half-life of 6 hours. Initial mass 48 g. Find remaining after 18 hours.
- Population grows 3% annually. Find population after 15 years if initial is 12,000.
- A city’s population triples every 50 years. Write an exponential function and find population after 100 years.