Algebra 101 - Unit 13

Lesson 84: Solving Real-World Problems with Nonlinear Systems

Nonlinear systems can model complex real-world scenarios in physics, engineering, biology, and economics. Understanding how to set up equations from word problems is key to finding solutions.

Strategy for Solving Problems

1. Read the problem carefully and identify the variables.
2. Translate relationships into equations (linear and nonlinear).
3. Choose a method (substitution, elimination, or graphing) to solve the system.
4. Interpret your solutions in the context of the problem.
5. Verify your solution satisfies all constraints.

Examples

Example 1 — Physics (Projectile Motion):

A projectile’s height is given by y = −4.9t² + 20t. A wall is located at y = 15. Find the times when the projectile hits the wall.

Solve: −4.9t² + 20t = 15 → 4.9t² − 20t + 15 = 0 → t = [20 ± √(400 − 294)]/9.8 → t ≈ 1.03 s, 2.97 s

Example 2 — Economics:

Revenue R(x) = 50x, Cost C(x) = 2x² + 10x + 200. Break-even occurs when R(x) = C(x): 50x = 2x² + 10x + 200 → 2x² − 40x + 200 = 0 → x² − 20x + 100 = 0 → x = 10 units

Example 3 — Geometry:

A circular fountain has radius 5 m. A walkway is represented by y = x + 2. Find intersection points: x² + y² = 25 → x² + (x + 2)² = 25 → x² + x² + 4x + 4 = 25 → 2x² + 4x − 21 = 0 → x = [−4 ± √(16 + 168)]/4 → x ≈ 2.28, −4.6 → corresponding y-values: y ≈ 4.28, −2.6

Practice Problems

1. A rocket’s height is h = −5t² + 30t. A platform is at h = 20. Find times when the rocket is at the platform.
2. A company’s profit P(x) = −x² + 15x − 20. Solve for break-even points.
3. Intersection of a circular garden x² + y² = 36 and a straight path y = x − 2.
4. A tank’s volume V = xyz with constraint x² + y² + z² = 36. Find maximum volume.
5. A parabolic bridge y = −x² + 10x intersects the line y = 8. Find x-coordinates of intersection.