1. Problem statement: Calculate areas A1, A2, A3, total area, and perimeter of the composite shape formed by three rectangles with given dimensions. 2. Calculate area A1: A1 is a rectangle of width 12 m and height 4 m. $$A_1 = \text{width} \times \text{height} = 12 \times 4 = 48\,m^2$$ 3. Calculate area A2: A2 is a rectangle of width 2 m and height 4 m. $$A_2 = 2 \times 4 = 8\,m^2$$ 4. Calculate area A3: A3 is a rectangle of width 12 m and height 8 m. $$A_3 = 12 \times 8 = 96\,m^2$$ 5. Calculate total area of the composite shape: The composite shape consists of A1, A2, and A3 combined without overlapping. $$\text{Total area} = A_1 + A_2 + A_3 = 48 + 8 + 96 = 152\,m^2$$ 6. Calculate the perimeter of the composite shape: - Left side height: 4 + 4 + 8 = 16 m (Verify question states 15 m total, so use given distances on shape: left side total height is 4 + 4 + 8 = 16 m) - Bottom width: 12 m - Width along top: 12 m Analyzing the perimeter step-by-step: - Bottom edge: 12 m - Right edge: height of bottom rectangle = 8 m - Top edge: 12 m - Left edges: sum up vertical segments: 4 + 4 + 8 = 16 m Since the shape is compound, perimeter segments = bottom (12) + right (8) + top (12) + left (16) = 12 + 8 + 12 + 16 = 48 m Final perimeter: $$P = 48\,m$$ --- 7. Problem statement: Analyze function $$y = -\frac{3}{2}(x+2)^2 - 4$$ 8. Find the axis of symmetry: The axis of symmetry formula for vertex form $$y = a(x-h)^2 + k$$ is $$x = h$$. Here, $$h = -2$$ Axis of symmetry: $$x = -2$$ 9. Find the vertex coordinates: Vertex is at $$(h, k) = (-2, -4)$$ 10. Find the y-intercept: Set $$x=0$$: $$y = -\frac{3}{2}(0+2)^2 - 4 = -\frac{3}{2} \times 4 - 4 = -6 - 4 = -10$$ Y-intercept: $$(0, -10)$$ 11. Express function in standard form $$y = ax^2 + bx + c$$: Expand the vertex form: $$y = -\frac{3}{2}(x+2)^2 - 4 = -\frac{3}{2}(x^2 + 4x + 4) - 4 = -\frac{3}{2}x^2 - 6x - 6 - 4$$ Simplify: $$y = -\frac{3}{2}x^2 - 6x - 10$$ 12. Calculate the discriminant $$\Delta = b^2 - 4ac$$: Here, $$a = -\frac{3}{2}, b = -6, c = -10$$ Calculate: $$\Delta = (-6)^2 - 4 \times \left(-\frac{3}{2}\right) \times (-10) = 36 - 4 \times \frac{3}{2} \times 10 = 36 - 60 = -24$$ Discriminant is negative, meaning no real roots. --- 13. Problem statement: Analyze rainfall data for PNG first 90 days. 14. Find the range: Range = max - min rainfall: Max = 280 mm, Min = 0 mm $$\text{Range} = 280 - 0 = 280\,mm$$ 15. Number of days with rainfall < 161 mm: Add days for intervals 0-40, 41-80, 81-120, 121-160: $$10 + 16 + 25 + 30 = 81\,\text{days}$$ 16. Number of days with rainfall between 45 mm and 200 mm: Include intervals 41-80 (partially), 81-120, 121-160, 161-200 (partially): From 45 to 200 mm includes 41-80 fully (16 days), 81-120 (25 days), 121-160 (30 days), and part of 161-200 (22 days). Assuming full intervals are included except 41-80 starts from 45 (>41) and 161-200 ends at 200 (includes all 22 days): Total: $$16 + 25 + 30 + 22 = 93\,\text{days}$$ (Note total days exceeds 90; so clarify intervals and adjust by partial) Since total days are 90, adjust calculations: - 0-40: 10 days - 41-80: 16 days - 81-120: 25 days - 121-160: 30 days - 161-200: 22 days - Sum: 10+16+25+30+22=103 exceeds total 90, likely overlapping counts or data error Treat all as full intervals; question expects summation of days where rainfall lies between 45 and 200 mm: Days with rainfall 41-80 =16 (include all), 81-120=25, 121-160=30, 161-200=22 From 45 mm, remove some from 41-80 interval (41-44 mm days unknown), assume entire 16 days included. So approximate: $$16 + 25 + 30 + 22 = 93$$ (acknowledge data inconsistency but answer as asked) 17. Most frequent rainfall measurement: Identify interval with maximum days: Maximum days = 30 (121-160 mm interval) 18. Days with rainfall > 200 mm: Sum days in 201-240 and 241-280 intervals: $$12 + 5 = 17\,\text{days}$$