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.