A rectangular area of 12 km² is represented by 1.5 cm × 0.5 cm rectangle on a map. The scale of the map is:
Correct Answer:
C. 1: 400000
📝 Detailed Explanation: Calculating Scale from Area
This problem requires finding the linear scale (the R.F.) when given the area on a map and the corresponding area on the ground. The key concept to remember is the relationship between linear scale and area scale.
Because area is a two-dimensional measurement (length × width), the area scale is the square of the linear scale (R.F.). To solve this, we must first find the area scale and then take its square root to find the R.F.
Step-by-Step Calculation
- Calculate Map Area:
Map Area = 1.5 cm × 0.5 cm = 0.75 cm² - Calculate Ground Area in the Same Units (cm²):
- First, convert km to cm: 1 km = 100,000 cm.
- Now, find the conversion for area: 1 km² = (100,000 cm)² = 10,000,000,000 cm² (or 10¹⁰ cm²).
- Ground Area = 12 km² = 12 × 10¹⁰ cm². - Determine the Area Scale:
Area Scale = Map Area / Ground Area
Area Scale = 0.75 cm² / (12 × 10¹⁰ cm²) = 1 / (16 × 10¹⁰) - Calculate the Linear Scale (R.F.):
R.F. = √(Area Scale)
R.F. = √[1 / (16 × 10¹⁰)] = 1 / [√(16) × √(10¹⁰)] = 1 / (4 × 10⁵) = 1 / 400,000 - Express as a Representative Fraction:
The scale of the map is 1 : 400,000.
⚗️ Formulas & Applications
The relationship between linear and area measurements on a map is fundamental.
Area Scale = (Linear Scale)² = (R.F.)²
R.F. = √ (Map Area / Ground Area)
Practical Example
A map shows a lake with an area of 5 cm². The true area of the lake is 5 km². What is the R.F. of the map?
- Step 1: Convert ground area to cm².
- 5 km² = 5 × (100,000 cm)² = 5 × 10¹⁰ cm²
- Step 2: Calculate the R.F.
- R.F. = √ (5 cm² / (5 × 10¹⁰ cm²))
- R.F. = √ (1 / 10¹⁰) = 1 / 10⁵ = 1 / 100,000
- Step 3: State the R.F.
- The scale is 1 : 100,000.
💡 Key Concepts for Students
- Area Scale is Squared: This is the most important takeaway. Forgetting to square the linear scale (or take the square root of the area scale) is a very common mistake.
- Unit Consistency is Crucial: Before you can calculate the area ratio, both the map area and ground area MUST be in the same units (e.g., cm² or m²).
- Large Number Conversions: Practice working with large numbers and scientific notation (like 10¹⁰) to avoid calculation errors with unit conversions for area.