Section 2.4: Applications of Quadratics

This section explores real-world problems modeled by quadratic functions, including projectile motion, optimization, and area problems.

Example 1: Projectile Motion

A ball is thrown upward with height \( h(t) = -16t^2 + 64t + 80 \). Find the maximum height and time it occurs.

Vertex formula: \( t = -\frac{b}{2a} = -\frac{64}{2(-16)} = 2 \) seconds

Maximum height: \( h(2) = -16(2)^2 + 64(2) + 80 = 144 \) ft

Example 2: Area Optimization

Find dimensions of a rectangle with perimeter 40 m that maximize area.

Let width = x → length = 20 - x

Area: A = x(20 - x) = -x^2 + 20x

Maximum at vertex: x = -b/(2a) = -20/(2(-1)) = 10

Length = 20 - 10 = 10 → square gives max area 100 m²

Practice Problems

  1. A ball is thrown with height \( h(t) = -16t^2 + 32t + 48 \). Find max height.
  2. Rectangle perimeter 50 m. Find dimensions for max area.
  3. Projectile motion: \( h(t) = -5t^2 + 20t + 15 \). Determine time to hit ground.
  4. Revenue problem: \( R(x) = -5x^2 + 150x \). Find x for max revenue.
  5. Area problem: Farmer fencing problem: maximize area of pen with 60 m fencing.
← Previous Next →