A farmer has 120 m of fencing to enclose a rectangular field that lies along a straight river. No fencing is needed along the river. The maximum area of the field that can be enclosed is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1800 m2
Let each side perpendicular to the river be x m. The fence covers these two sides and the side parallel to the river, so that side is 120 − 2x m, with 0 < x < 60. The area is
\(A = x(120 - 2x)\)
\(A = 120x - 2x^2\)
Setting the derivative equal to zero:
\( \displaystyle \frac{dA}{dx} = 120 - 4x \)
\(120 - 4x = 0\)
\(x = 30\)
Since \(\frac{d^2A}{dx^2} = -4 \lt 0\), the area is a maximum at x = 30. The sides are 30 m and 120 − 60 = 60 m:
\(A = 30 \times 60 = 1800\) m2
Why the other options are incorrect
Option (2) 900 m2 — Incorrect. This is a 30 m by 30 m square, the best field when all four sides are fenced; here the river side needs no fence.
Option (3) 3600 m2 — Incorrect. A 60 m by 60 m field needs 60 + 60 + 60 = 180 m of fencing, more than 120 m.
Option (4) 1600 m2 — Incorrect. This is the field with three equal fenced sides of 40 m; A(40) = 40 × 40 = 1600, less than A(30) = 1800.