Section 6.5: Review & Assessment
This section reviews all exponential function topics and provides assessment problems to test understanding of growth, decay, graphing, equations, and applications.
Example 1: Growth Problem Review
The population of a town grows according to \( P(t) = 8000 \cdot 1.04^t \). Find the population after 5 years.
Step 1: Substitute t = 5 → \( P(5) = 8000 \cdot 1.04^5 \)
Step 2: Calculate → \( P(5) \approx 8000 \cdot 1.2167 \approx 9734 \)
Example 2: Compound Interest Review
Invest $2500 at 3% annual interest, compounded monthly. Find the amount after 4 years.
Step 1: \( A = 2500(1 + 0.03/12)^{12 \cdot 4} \)
Step 2: \( A \approx 2500 \cdot (1.0025)^{48} \approx 2500 \cdot 1.1275 \approx 2819 \)
Assessment Problems
- Population doubles every 6 years. Current population = 5000. Find population after 18 years.
- Radioactive decay: \( N(t) = 200(1/2)^{t/3} \). Find N after 9 years.
- Graph \( f(x) = 2 \cdot 3^x \) and describe its end behavior.
- Invest $1000 at 4% compounded quarterly. Find amount after 10 years.
- Find the time it takes for a $500 investment to grow to $800 at 5% annual compounded monthly.