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Question
There is a rectangular farm with length `(2a^2 + 3b^2)` metre and breadth `(a^2 + b^2)` metre. The farmer used a square shaped plot of the farm to build a house. The side of the plot was `(a^2 - b^2)` metre.
What is the area of the remaining part of the farm?
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Solution
Lenght of the rectangular farm = (2a2 + 3b2) m
Breadth of the rectangular farm = (a2 + b2) m
The total area of the farm = Length of the rectangular farm × Breadth of the rectangular farm
= (2a2 + 3b2) × (a2 + b2)
= 2a2(a2 + b2) + 3b2(a2 + b2)
= 2a4 + 2a2b2 +3a2b2 + 3b4
= (2a4 + 5a2b2 + 3b2) sq.meter
Side of the square plot = (a2 − b2) meter.
Area of the square plot = (side)2
= (a2 − b2)2
= a − 2a2b2 + b4
Area of the remaining part of the farm = Total area of the farm − Area of the square plot
= (2a4 + 5a2b2 + 3b4) - (a4 − 2a2b2 + b4)
= 2a4 + 5a2b2 + 3b4 − a4 − 2a2b2 + b4
= 2a4 − a4 + 5a2b2 + 2a2b2 + 3b4 − b4
= a4 + 7a2b2 + 2b4
Thus, the area of the remaining part of the farm is (a4 + 7a2b2 + 2b4) sq. meter.
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