Question

There is a square field with each side 500 metres long. It has a compound wall along its perimeter. At one of its corners, a triangular area of the field is to be cordoned off by erecting a straight line fence. The compound wall and the fence will form its borders. If the length of the fence is 100 metres, what is the maximum area in square metres that can be cordoned off?

• A. 5,000
• B. 10,000
• C. 2,500
• D. 20,000

Hint:

The two walls meet at a right angle, so the fence is the hypotenuse of a right triangle. Maximise half the product of the legs for a fixed hypotenuse.

Answer 1

The Correct Option is C

Step 1: Set up the triangle:
The two walls at the corner meet at \(90^\circ\). Let the legs along the walls be \(a\) and \(b\). The fence is the hypotenuse, so \(a^2 + b^2 = 100^2 = 10000\).

Step 2: Maximise the area:
Area \(= \frac{1}{2}ab\). Since \(a^2 + b^2 \ge 2ab\), we get \(ab \le 5000\), so area \(\le 2500\). Equality holds when \(a = b = 50\sqrt{2} \approx 70.7\) m, which is far less than 500 m, so it fits inside the field.

Step 3: Check options:
The maximum area is 2,500 sq. m. The values 5,000, 10,000 and 20,000 are too large (5,000 would be \(ab\), not half of it).

Final Answer:
The correct option is (3): 2,500