Quadratic Real-World Applications

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📊 Part 1: Understanding Quadratic Applications

1. A farmer has 80 metres of fencing to create a rectangular paddock against a river (no fence needed along the river). If the width is w metres, which expression represents the area?

A = w(80 - w)

A = w(80 - 2w)

A = 2w(80 - w)

A = w(40 - w)

2. A projectile is fired upward. Its height h (in metres) after t seconds is given by h = -5t² + 20t + 15. When does the projectile hit the ground?

t = 3 seconds

t = 4 seconds

t = 5 seconds

t = 6 seconds

3. A shop's daily profit P (in dollars) is modelled by P = -2x² + 120x - 1000, where x is the number of items sold. What does the vertex of this parabola represent?

The minimum profit

The maximum profit

The break-even point

The total cost

4. Which real-world situations can be modelled using quadratic functions? (Select all that apply)

The path of a basketball shot

The area of a rectangle with fixed perimeter

Simple interest calculations

The height of a bouncing ball over time

🔧 Part 2: Problem Solving

5. A rectangular garden has a perimeter of 60 metres. Let the length be l metres and width be w metres.

a) Write an expression for the area A in terms of w only.

b) What width gives the maximum area? Show your working.

c) What is the maximum area possible?

6. A ball is thrown from a cliff 25 metres high. Its height h (metres) after t seconds is given by h = -5t² + 10t + 25.

a) Factorise the expression -5t² + 10t + 25.

b) When does the ball reach its maximum height?

c) How long does it take for the ball to hit the ground? Show your working.

💡 Part 3: Digital Tools & Analysis

7. Using graphing software (GeoGebra or Desmos), graph the function A = w(30 - w) for the garden problem above.

a) What are the x-intercepts of this graph?

x-intercepts: w = _______ and w = _______

b) What is the vertex of this parabola?

Vertex: (_______, _______)

c) Explain what the vertex represents in the context of the garden problem.

8. A company's profit function is P = -x² + 40x - 300, where x is the number of products sold (in hundreds).

a) Use digital tools to find when the company breaks even (P = 0).

Break-even points: x = _______ and x = _______ (hundreds of products)

b) What number of products should be sold to maximise profit?

9. Extension: Describe a real-world situation from your own experience that could be modelled using a quadratic function. Explain why it would create a parabolic relationship.