The area of the plan of an old survey plotted to a scale of 15 m to 1 cm now measures 80·2 cm² as found by a planimeter. The plan is found to be shrunk, so that a line originally 10 cm long now measures 9·8 cm. The true area of the survey will be nearly
- A81·5 m²
- B83·5 m²Correct
- C85·5 m²
- D87·5 m²
Solution
Concept
Correction of area for shrinkage of the plan
Formula
True area = measured area × (L / L′)²
Step-by-step solution
1. Shrinkage factor = 10 / 9.8. 2. True area = 80.2 × (10/9.8)² = 80.2 × 1.0412 = 83.5 (in the same units as the measured area).
Explanation
Areas scale with the square of the linear shrinkage, so the measured area is multiplied by the square of the ratio of original to present length.
Why the other options are wrong
81.5, 85.5 and 87.5 do not equal 80.2 × (10/9.8)².
Shortcut
Corrected area = measured × (original/present)².
Common mistake
Using the ratio without squaring (gives 81.8).
Correct answer: B) 83·5 m²