The cost of fencing a rectangular plot is $200$ per ft along one side, and $100$ per ft along the three other sides. If the area of the rectangular plot is $60000$ sq. ft, then the lowest possible cost of fencing all four sides, in INR, is

We call the dimensions of the rectangular plot Length (L) and Breadth (B).

Given information:

Area = L × B = 60000 sq. ft

One side costs ₹200 per ft

The other three sides cost ₹100 per ft each

Since we want to minimize cost, we say one of the lengths costs ₹200 per ft.

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For a rectangle, we have:

Two lengths and Two breadths

One length costs ₹200 per ft → Cost = 200L

Other length costs ₹100 per ft → Cost = 100L

Two breadths cost ₹100 per ft each → Cost = 100B + 100B = 200B

Total Cost = 200L + 100L + 200B = 300L + 200B

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We need to minimize: Cost = 300L + 200B

Subject to the constraint: L × B = 60000

From the area constraint: $B = \frac{60000}{L}$

Substituting this into our cost function:

$\text{Cost} = 300L + 200 \times \frac{60000}{L} = 300L + \frac{12000000}{L}$

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When we have an expression like $aX + \frac{b}{X}$, the minimum occurs when both terms are equal.

For our cost function: $300L + \frac{12000000}{L}$

The minimum occurs when: $300L = \frac{12000000}{L}$

$300L^2 = 12000000$

$L^2 = 40000$

$L = 200$ ft

Therefore: $B = \frac{60000}{200} = 300$ ft

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With L = 200 ft and B = 300 ft:

One length at ₹200: 200 × 200 = ₹40000

Other length at ₹100: 100 × 200 = ₹20000

Two breadths at ₹100 each: 2 × 100 × 300 = ₹60000

Total minimum cost = 40000 + 20000 + 60000 = ₹120000

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The AM-GM inequality tells us that for positive numbers:

$\frac{300L + \frac{12000000}{L}}{2} \geq \sqrt{300L \times \frac{12000000}{L}} = \sqrt{3600000000} = 60000$

This means: $300L + \frac{12000000}{L} \geq 120000$

The equality (and thus minimum) occurs exactly when $300L = \frac{12000000}{L}$, which gives us L = 200 and B = 300.

Therefore, the lowest possible cost of fencing is ₹120000.