Lecture Notes: Quadratic Word Problems

Introduction

• Mr. Robinson welcomes viewers to another math video.
• The focus is on quadratic equations and word problems.

Series Overview

• Previous topics include graphing quadratic functions.
• Current focus: Quadratic word problems, not directly graphing but understanding their meaning through graphs.

Key Concepts of Word Problems

• Finding x (or t): Often used interchangeably with y (h of t in this context).
• Problems will frequently involve maximum points related to graphs, particularly upside-down parabolas due to gravity.
• The standard model incorporates gravity:
  • Model form: h(t) = -16t^2 + vt + h_0
  • Where -16 represents gravity's effect in feet/second.

Example 1: Jason's Jump

Initial Function

• Height function: h(t) = -16t^2 + 16t + 480
  • t = time (seconds), h = height (feet).

Questions and Solutions

• Maximum Height:
  • Use the vertex formula: t = -b/(2a)
    • For this equation, t = -16/(2 * -16) = 0.5 seconds
  • Maximum Height Calculation:
    Substitute into function: h(0.5) = -16(0.5)^2 + 16(0.5) + 480
    Results in a maximum height of 484 feet.
• Time to Hit Water:
  • Set height equation to zero:
    -16t^2 + 16t + 480 = 0
  • Factor or use quadratic formula to find t values:
    • Positive solution: 6 seconds
    • Negative solution deemed irrelevant.*

Example 2: Toy Rocket Launch

Initial Function

• Height function: h(t) = -16t^2 + 128t

Questions and Solutions

• Time to Hit Ground:
  • Set to zero, solve for t:
    Results in 8 seconds.
• Height at 112 Feet:
  • Set the equation to 112:
    • Two solutions: 1 second (upward) and 7 seconds (downward).
• Maximum Height:
  • Use -b/(2a) to find t = 4 seconds, substitute to find height = 256 feet.

Example 3: Rocket from Cliff

Initial Function

• Height function: h(t) = -16t^2 + 116t + 101

Questions and Solutions

• Time to Hit Ground:
  • Requires using the quadratic formula for precision leading to 8 seconds.
• Maximum Height:
  • Use vertex method or quadratic formula to find the maximum height.

Example 4: Grappling Hook

Initial Function

• Height function: h(t) = -16t^2 + 32t + 5

Questions and Solutions

• Maximum Height:
  • Calculated as 21 feet; can reach the 20-foot ledge.

Example 5: Jumping Basketball

Initial function

• Height function: h(t) = -16t^2 + 12t

Questions and Solutions

• Maximum Height: Calculate and find maximum height of 2.25 feet; cannot dunk (needs 2.5 feet).

Conclusion

• Importance of understanding graphical interpretation of quadratic equations in word problems.
• Methods discussed: factoring, using quadratic formula, completing the square, and vertex formula.