An agricultural field is in the form of a rectangle having length X₁ meters and breadth X₂ meters (X₁ and X₂ are variable). If X₁ + X₂ = 40 meters, then the area of the agricultural field will not exceed which one of the following values ?

(a) 400 sq m ✓ UPSC's answer
(b) 300 sq m
(c) 200 sq m
(d) 80 sq m

Why the answer is (a)

• The area of the rectangular field is given by the product of its length and breadth, Area = X₁ * X₂.

• The problem states that the sum of the length and breadth is constant: X₁ + X₂ = 40.

• To find the maximum possible area, we use the property that for a fixed sum, the product of two numbers is maximized when the numbers are equal.

• Therefore, the maximum area occurs when X₁ = X₂ = 40 / 2 = 20 meters.

• Calculating the area at this point: 20 * 20 = 400 square meters.

• Thus, the area of the field will not exceed 400 sq m, which corresponds to option (a).

Why the other options are wrong

(b) 300 sq m: 300 sq m is less than the maximum possible area of 400 sq m, so the area can exceed this value.
(c) 200 sq m: 200 sq m is significantly lower than the maximum area of 400 sq m, making it an incorrect upper bound.
(d) 80 sq m: 80 sq m is far below the maximum area of 400 sq m and does not represent the upper limit of the field's area.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2016, held on 7 August 2016. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.