Section 8.6: Applications of Quadratics
Quadratic functions model real-life situations such as projectile motion, area problems, and revenue maximization.
Example 1: Projectile Motion
A ball is thrown upward with height \( h(t) = -5t^2 + 20t + 1 \) meters after \( t \) seconds. Find the maximum height and time it occurs.
Vertex formula: \( t = -\frac{b}{2a} = -\frac{20}{2(-5)} = 2 \) seconds.
Maximum height: \( h(2) = -5(2)^2 + 20(2) + 1 = -20 + 40 + 1 = 21 \) meters.
Example 2: Area Problem
A rectangular garden has a perimeter of 40 m. Express area \( A \) as a quadratic function of length \( x \) and find the maximum area.
Width: \( w = 20 - x \), Area: \( A = x(20 - x) = -x^2 + 20x \)
Vertex: \( x = -\frac{b}{2a} = -\frac{20}{2(-1)} = 10 \) m
Maximum area: \( A = 10 \cdot 10 = 100 \) m²
Practice Problems
- A ball is thrown: \( h(t) = -4t^2 + 16t + 2 \). Find max height and time.
- Rectangle with perimeter 50 m: express area as quadratic, find max area.
- Revenue problem: \( R(x) = -5x^2 + 100x \). Find number of items sold for max revenue.
- Projectile: \( h(t) = -6t^2 + 24t + 3 \). Find when ball hits ground.
- Area of a fenced garden: sides differ by 2 m, perimeter 24 m. Max area?