Scale Drawings and Maps — Solutions

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1. Map distance to actual distance

   1. 4.5 × 10 000 = 45 000 cm = 450 m
   2. 7.2 × 50 000 = 360 000 cm = 3 600 m = 3.6 km
   3. 3 × 200 = 600 cm = 6 m
   4. 12.5 × 100 000 = 1 250 000 cm = 12 500 m = 12.5 km
   5. 0.8 cm × 25 000 = 20 000 cm = 200 m

2. Actual distance to map distance

   1. 3 km = 300 000 cm. Map = 300 000 ÷ 50 000 = 6 cm
   2. 14 m = 1400 cm. Map = 1400 ÷ 200 = 7 cm
   3. 4.5 km = 450 000 cm. Map = 450 000 ÷ 100 000 = 4.5 cm
   4. 750 m = 75 000 cm. Map = 75 000 ÷ 10 000 = 7.5 cm
   5. 2 km = 200 000 cm. Map = 200 000 ÷ 25 000 = 8 cm

3. Find the scale

   1. 2 cm : 500 m = 2 cm : 50 000 cm = 1 : 25 000. Scale = 1 : 25 000
   2. 5 cm : 1 km = 5 cm : 100 000 cm = 1 : 20 000. Scale = 1 : 20 000
   3. 4 mm : 2 m = 0.4 cm : 200 cm = 1 : 500. Scale = 1 : 500
   4. 8 cm : 40 m = 8 cm : 4000 cm = 1 : 500. Scale = 1 : 500
   5. 3 cm : 1.5 km = 3 cm : 150 000 cm = 1 : 50 000. Scale = 1 : 50 000

4. Floor plan (scale 1 : 100)

   1. 5.5 × 100 = 550 cm = 5.5 m; 4 × 100 = 400 cm = 4 m. Actual dimensions: 5.5 m × 4 m
   2. 3.6 m ÷ 100 = 0.036 m = 3.6 cm; 2.8 m ÷ 100 = 2.8 cm. Plan: 3.6 cm × 2.8 cm
   3. 5.5 × 4 = 22 m²
   4. Plan area = 5.5 × 4 = 22 cm². Real area = 22 m² = 22 × 10 000 cm² = 220 000 cm². Ratio = 220 000/22 = 10 000 = 100². The plan area is 10 000 times smaller.
   5. Real area = 18 × 12 = 216 m². Plan area = 216/100² = 216/10 000 m². In cm² on plan: 18 cm × 12 cm = 216 cm²

5. Topographic map (scale 1 : 25 000)

   1. 14.8 × 25 000 = 370 000 cm = 3700 m = 3.7 km
   2. Actual dimensions: 3.2 × 250 m = 800 m by 1.4 × 250 m = 350 m. Area = 800 × 350 = 280 000 m² = 0.28 km²
   3. 22.5 km = 2 250 000 cm. Map distance = 2 250 000 ÷ 25 000 = 90 cm
   4. Area scale factor = 25 000² = 625 000 000. So 1 cm² on map = 625 000 000 cm² = 62 500 m². The student is incorrect; 1 cm² represents 62 500 m², not 625 m².

6. Engineering drawing (scale 2 : 1, drawing is larger)

   1. Actual = 4.8 ÷ 2 = 2.4 cm
   2. Actual diameter = 1.5 ÷ 2 = 0.75 cm
   3. Actual dimensions: 3 cm × 1.5 cm. Actual area = 3 × 1.5 = 4.5 cm²
   4. Drawing at a scale larger than 1:1 makes small components easier to see, measure, and annotate with precise dimensions, reducing errors during manufacture.

7. Suburban park planning (scale 1 : 2000)

   1. 7.5 × 2000 = 15 000 cm = 150 m; 4 × 2000 = 8000 cm = 80 m. Actual: 150 m × 80 m
   2. Area = 150 × 80 = 12 000 m² = 1.2 ha
   3. 8.6 × 2000 = 17 200 cm = 172 m
   4. Cost = 12 000 × $12 = $144 000
   5. Perimeter = 2(150+80) = 460 m. Cost = 460 × $85 = $39 100

8. Cross-country hike (scale 1 : 50 000)

   1. Leg 1: 6.4 × 50 000 = 320 000 cm = 3.2 km. Leg 2: 9.2 × 50 000 = 460 000 cm = 4.6 km. Leg 3: 11.3 × 50 000 = 565 000 cm = 5.65 km
   2. Total = 3.2 + 4.6 + 5.65 = 13.45 km
   3. Time = 13.45 ÷ 4 = 3.3625 h = 3 h 21.75 min ≈ 3 hours 22 minutes
   4. If Leg 1 (N) and Leg 2 (E) are perpendicular: hypotenuse = √(3.2² + 4.6²) = √(10.24 + 21.16) = √31.4 ≈ 5.603 km. Leg 3 = 5.65 km. These are close but not exactly equal — the map distances may include slight curves, or the diagonal is not a perfect straight line. The values are approximately consistent.

9. Scale model house (scale 1 : 40)

   1. Length = 36 × 40 = 1440 cm = 14.4 m; Width = 22 × 40 = 880 cm = 8.8 m
   2. Roof height = 18 × 40 = 720 cm = 7.2 m
   3. Window: 1.5 × 40 = 60 cm = 0.6 m wide; 2 × 40 = 80 cm = 0.8 m tall. Actual: 0.6 m × 0.8 m
   4. Floor area = 14.4 × 8.8 = 126.72 m²
   5. Cost = 126.72 × $45 = $5 702.40

10. Comparing two maps

    1. 3.5 km = 350 000 cm. On Map A (1:10 000): 350 000 ÷ 10 000 = 35 cm
    2. On Map B (1:50 000): 350 000 ÷ 50 000 = 7 cm
    3. Map A shows more detail. At a smaller scale ratio (1:10 000), features are drawn larger, so more detail can be seen and labelled. Map B (1:50 000) covers a larger area but with less detail per unit area.
    4. Area scale factor = 10 000² = 10&sup8;. Forest area = 48 cm² × 10&sup8; cm²/cm² = 48 × 10&sup8; cm² = 48 × 10&sup8; ÷ 10¹0; km² = 4.8 km²
    5. On Map B: area scale factor = 50 000² = 25 × 10&sup8;. Map B area = 4.8 km² = 4.8 × 10¹0; cm². Map B forest = 4.8 × 10¹0; ÷ (25 × 10&sup8;) = 4.8/25 × 10² cm² = 1.92 cm². 1.92 cm². (Alternatively: Map B = 48 × (10 000/50 000)² = 48 × (1/5)² = 48/25 = 1.92 cm².)