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.