Section 8.9: Real-Life Applications
Quadratic functions are not just abstract—they model many real-life situations such as projectile motion, profit and revenue optimization, and geometric problems.
Example 1: Projectile Motion
A ball is thrown upward with height modeled by \( h(t) = -5t^2 + 20t + 1 \), where \( h \) is in meters and \( t \) is in seconds.
- Find the maximum height of the ball.
- Find when the ball hits the ground.
Vertex formula: \( t = -\frac{b}{2a} = -\frac{20}{2(-5)} = 2 \) seconds.
Max height: \( h(2) = -5(4) + 20(2) + 1 = -20 + 40 + 1 = 21 \) m.
To find when it hits the ground: solve \( -5t^2 + 20t + 1 = 0 \). Using quadratic formula gives \( t \approx 0.05 \) s (initial) and \( t \approx 4.0 \) s (impact).
Example 2: Profit Optimization
A company models profit with \( P(x) = -2x^2 + 40x - 100 \), where \( x \) is the number of units produced.
- Find the number of units that maximizes profit.
- Find the maximum profit.
Vertex: \( x = -\frac{b}{2a} = -\frac{40}{2(-2)} = 10 \).
Max profit: \( P(10) = -200 + 400 - 100 = 100 \).
Example 3: Geometry Application
The area of a rectangle is modeled as \( A(x) = x(20-x) \). Find the value of \( x \) that maximizes area.
\( A(x) = -x^2 + 20x \), a downward parabola.
Vertex at \( x = -\frac{b}{2a} = -\frac{20}{2(-1)} = 10 \).
So maximum area occurs when \( x = 10 \), giving a square of side 10.