Learn how to translate real-world scenarios into quadratic and exponential equations and solve them systematically.
Example: A ball is thrown upward from a height of 5 feet with an initial velocity of 48 ft/s. Its height h after t seconds is given by h = -16t² + 48t + 5. What is the maximum height reached?
Solution: The maximum occurs at the vertex. t = -b/(2a) = -48/(2×-16) = 1.5 seconds. Then h = -16(1.5)² + 48(1.5) + 5 = 41 feet.
Example: A bacteria culture starts with 500 bacteria and doubles every hour. How many bacteria will there be after 6 hours?
Solution: Exponential growth model: N = 500 × 2^t. After 6 hours: N = 500 × 2^6 = 500 × 64 = 32,000 bacteria.
| Keywords | Model Type |
|---|---|
| "maximum," "minimum," "vertex," "parabola" | Quadratic |
| "doubles," "triples," "half-life," "grows by percentage" | Exponential |
| "constant rate" or "linear change" | Linear |
A car's value depreciates exponentially over time. The car was originally purchased for $25,000 and was worth $20,000 after 1 year. What will its value be after 3 years? (Round to the nearest dollar.)
Follow these steps:
Step-by-Step Solution:
1. Find decay factor: 20,000 = 25,000 × r^1 → r = 0.8
2. Model: V = 25,000 × (0.8)^t
3. After 3 years: V = 25,000 × (0.8)^3 = 25,000 × 0.512 = $12,800
The correct answer is A) $12,800.