The length of a line originally 100 mm long on a map plotted to a scale of 1/1000, was found to be 96 mm due to shrinkage of the map. The map prepared using a tape of length 20 m was later found to be actually 20.03 m. If a certain area on the map, measured using a planimeter, is 282 mm², what is the correct area on the ground?
- A237 m²
- B307 m²Correct
- C347 m²
- D397 m²
Solution
Concept
Area is corrected by the square of the shrinkage factor and the square of the tape length ratio
Formula
True area = (Measured area / SF2) × (Scale)2 × (L'/L)2
Step-by-step solution
1. Shrinkage factor SF = 96/100 = 0.96 2. Original map area = 282 / 0.962 = 282 / 0.9216 = 305.99 mm² 3. Scale 1:1000, so 1 mm = 1 m and 1 mm² = 1 m². Ground area = 305.99 m² 4. Tape correction factor = (20.03/20)2 = 1.003 5. Correct area = 305.99 × 1.003 = 306.9 ≈ 307 m²
Explanation
First remove the map shrinkage by dividing by the square of the shrinkage factor. Then convert to ground area using the scale. The tape was longer than marked, so the true area is larger. Multiply by (20.03/20)2 to get about 307 m².
Why the other options are wrong
A: multiplies by the shrinkage factor instead of dividing. C and D: wrong use of the correction factors.
Shortcut
Correct area = 282 × (20.03/20)2 / 0.962 ≈ 307 m²
Common mistake
Using the first power of the factors instead of the square for area.
Correct answer: B) 307 m²