Applications of Non-Linear Equations: Fresh Take

  • Build and use equations and formulas that aren’t straight lines to solve real-life problems.

Quadratic Applications

The Main Idea

  • Projectile Motion: The curved path of an object launched into the air and affected by gravity
  • Quadratic Model: [latex]h = at^2 + bt + c[/latex], where:
    • [latex]h[/latex] is height
    • [latex]t[/latex] is time
    • [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are constants
  • Parabolic Trajectory: The shape of the path followed by a projectile
  • Real-world appliations
    • Satellite launches
    • Rocket science
    • Sports (e.g., basketball shots, football passes)
    • Ballistics
    • GPS-enabled tracking projectiles in law enforcement

Problem-Solving Strategy

  1. Identify the given quadratic equation
  2. Determine what information is required (e.g., time to hit ground, maximum height)
  3. Set up the equation based on the question:
    • For ground impact: Set height to 0
    • For specific height: Set height to the given value
  4. Solve the resulting quadratic equation
  5. Interpret the results in the context of the problem

 

You can view the transcript for “Ex: Quadratic Function Application Using Formulas – Rocket Launch” here (opens in new window).

Areas and Volumes

The Main Idea

  • Geometric Formulas:
    • Area formulas (2D shapes)
    • Volume formulas (3D shapes)
    • Perimeter and surface area
  • Variable Relationships:
    • Linear (e.g., perimeter of a rectangle)
    • Quadratic (e.g., area of a circle)
    • Cubic (e.g., volume of a sphere)
  • Problem-Solving Approach:
    • Identify relevant shapes
    • Determine given and unknown variables
    • Select appropriate formulas
    • Set up and solve equations
    • Interpret results in context
Shape Formula Variables
Rectangle (Area) [latex]A = lw[/latex] [latex]l[/latex] = length, [latex]w[/latex] = width
Circle (Area) [latex]A = \pi r^2[/latex] [latex]r[/latex] = radius
Triangle (Area) [latex]A = \frac{1}{2}bh[/latex] [latex]b[/latex] = base, [latex]h[/latex] = height
Rectangular Prism (Volume) [latex]V = lwh[/latex] [latex]l[/latex] = length, [latex]w[/latex] = width, [latex]h[/latex] = height
Sphere (Volume) [latex]V = \frac{4}{3}\pi r^3[/latex] [latex]r[/latex] = radius