Mathematical Methods and Problem Solving

Nov 25, 2024

Lecture Notes: Geometry and Calculations

Circles and Square Problem

  • Given: 3 circles inside a square, radius of each circle = 24mm
  • Objective: Find the area of the rectangle
  • Width Calculation:
    • 4 radiuses along the width = 4 x 24 = 96mm
  • Height Calculation:
    • Overlapping radiuses vertically; use equilateral triangle
    • Triangle sides = 48mm (2 x radius)
    • Use trigonometry (SOHCAHTOA) to find height
    • Height = 48 x sin(60°) = 41.5mm
    • Total height = 41.5 + 24 + 24 = 89.569mm
  • Area of Rectangle:
    • Area = Width x Height = 96 x 89.569 = 8,598.624 mm²
    • Rounded to 3 significant figures = 8600 mm²

Density and Bounds

  • Objective: Calculate density with bounds
  • Formula: Density = Mass / Volume
  • Mass and Volume Bounds:
    • Mass measured at 1970, correct to nearest 5
    • Volume calculated from given dimensions with error intervals
  • Error Intervals for Dimensions:
    • Length: 13.2 (13.15 to 13.25)
    • Width: 16.0 (15.95 to 16.05)
    • Height: 21.7 (21.65 to 21.75)
  • Mass Bounds:
    • Upper: 1972.5, Lower: 1967.5
  • Volume Bounds Calculation:
    • Upper volume: Product of upper bounds
    • Lower volume: Product of lower bounds
  • Density Bounds Calculation:
    • Upper Density = Upper Mass / Lower Volume
    • Lower Density = Lower Mass / Upper Volume
  • Final Density: Approximated to 0.43 using bounds

Similar Shapes

  • Given: Similar triangles with tangent function
  • Objective: Solve for scale factor
  • Scale Factor Calculation:
    • Equate ratios of corresponding sides
    • Use algebra to solve for unknowns

Probability and Ratios

  • Given: Probability of colored counters
  • Objective: Determine number of counters
  • Approach:
    • Use given ratios and changes
    • Algebraic manipulation to solve for counters

Bearings Calculation

  • Objective: Calculate bearing from given points
  • Approach:
    • Use angles and known rules (co-interior, angles around a point)
    • Use cosine rule for angles and sides
    • Calculate bearing as a sum of angles

3D Trigonometry

  • Objective: Determine angles in a 3D space
  • Approach:
    • Break problem into right triangle calculations
    • Use trigonometry (SOHCAHTOA) and Pythagorean theorem

Circle Equations and Tangents

  • Objective: Find equation of tangent to a circle
  • Approach:
    • Determine gradient of radius
    • Use negative reciprocal for tangent gradient
    • Calculate equation using point and gradient

Quadratic Simultaneous Equations

  • Objective: Solve quadratic equations simultaneously
  • Approach:
    • Rearrange one equation and substitute into another
    • Solve resulting quadratic
    • Use solutions to find values for variables

Vectors and Ratios

  • Objective: Solve for vector ratios
  • Approach:
    • Use given vectors and ratios
    • Algebraic manipulation to solve for unknown ratios

Trigonometry and Exact Values

  • Objective: Prove trigonometric equation
  • Approach:
    • Use cosine rule and known values
    • Simplify and rearrange equation to match given proof

These notes provide a summary of important mathematical methods and problem-solving approaches discussed in the lecture.