Section 3.5: Applications & Review
This section demonstrates real-world applications of polynomial functions and provides a review of key concepts from the chapter.
Example 1: Polynomial Modeling
A ball is thrown and its height in meters is modeled by \( h(t) = -2t^3 + 9t^2 + 12t \), where t is in seconds. Find the time when the ball hits the ground.
Step 1: Set h(t) = 0 → -2t^3 + 9t^2 + 12t = 0
Step 2: Factor: t(-2t^2 + 9t + 12) = 0 → t = 0 or solve quadratic
Step 3: Quadratic: -2t^2 + 9t + 12 = 0 → t = -1.5 (discard) or t = 4
Step 4: Ball hits ground at t = 4 seconds
Example 2: Profit Function
The profit P(x) in dollars for selling x units is given by \( P(x) = -5x^3 + 150x^2 - 1200x + 4000 \). Determine the number of units to maximize profit.
Step 1: Take derivative: P'(x) = -15x^2 + 300x - 1200
Step 2: Set P'(x) = 0: -15x^2 + 300x - 1200 = 0 → x^2 - 20x + 80 = 0
Step 3: Solve quadratic: x = 10 ± 2√5 → maximum occurs at x ≈ 5.52 or x ≈ 14.48
Practice Problems
- A cubic function models the volume of water in a tank. Find when the tank empties.
- Find the maximum height of a projectile given by h(t) = -t^3 + 6t^2 + 9t.
- Use synthetic division to evaluate P(2) for P(x) = x^4 - 3x^3 + 2x - 5.
- Determine zeros of profit function P(x) = -2x^3 + 12x^2 + 18x.
- Factor and find zeros of R(x) = x^3 - 4x^2 - 7x + 10.