Lesson 1.7

Real-World Models

Math doesn't just live on paper. We use quadratics to launch rockets, maximize business profits, and build bridges.

Introduction

Mathematics is the language of the physical world. In this final lesson of the chapter, we apply everything we've learned to solve real-world problems involving projectile motion, area maximization, and revenue optimization.

Past Knowledge

You have mastered Vertex Form (for max/min), Standard Form (for general equations), and Intercept Form (for roots).

Today's Goal

We translate word problems into quadratic equations. We ask: "Does this problem want the vertex (maximum) or the intercepts (landing point)?"

Future Success

This is the foundation of optimization in Calculus. Businesses pay millions for these exact calculations.

Key Concepts

The Dictionary of Quadratics

Translating English to Math is the hardest part. Here is your cheat sheet:

1
"Find the Maximum Height / Max Profit"Find the Vertex (y-value).
2
"When does it land? / Break-Even Points"Find the x-intercepts (Roots).
3
"Initial Height / Starting Cost"Find the y-intercept (c).

Projectile Model (Imperial):

t = time, v0 = initial velocity, h0 = initial height.

The flight path of a projectile.

Worked Examples

Example 1: Projectile Motion

Physics

A rocket is launched from 80 ft. Initial velocity is 64 ft/s.
Model: .
1. When does it reach max height?
2. What is the max height?

Find 'When' (x-value of Vertex)

Find 'Height' (y-value of Vertex)

Plug t=2 back in:

Example 2: Area Maximization

Geometry

Farmer Joe has 100m of fence to build a rectangular pen. What dimensions give the maximum area?

Build Equation

Perimeter .
Area .

Find Vertex (Midpoint)

Using Intercept Form : Intercepts at 0 and 50. Midpoint is 25.
So Width = 25m. Length = 50 - 25 = 25m.
Max Area = 25 * 25 = 625 sq m. (A Square!)

Example 3: Minimizing Cost

Business

Daily cost for a factory is . Find the production level that minimizes cost.

Find Min 'x' (Vertex)

Production Level: 20 units

Find Min Cost

Min Cost: $100

Common Pitfalls

Answering the Wrong Question

If the question asks "How high?", give the y-value. If it asks "When?", give the x-value. Always double check what variable you found!

Negative Time

When solving for landing time, you might get and . Reject the negative time! Math doesn't know physics, but you do.

Real-Life Applications

Revenue Optimization: A concert venue sells tickets for $50 and fills 2000 seats. For every $1 increase in price, they sell 50 fewer tickets.
Revenue = (Price)(Sales) = . This is a quadratic equation! Finding the vertex tells the venue exactly how much to raise the price to make the most money possible.