Question

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

• A. 90000
• B. 160000
• C. 120000
• D. 100000

Answer 1

The Correct Option is C

Let's assume total cost = \(200L+100B+100L+300L+200B……(1)\)

Area of rectangle = \(L \times B=600 …..(2)\)

We know that, \(AM> GM\)

\(=300L+200B>2  \times60000\)

Total cost is \(> 120000\)

∴ The lowest possible cost of fencing all four sides in INR is 120000.

Answer 2

Let the length and breadth be \(x\) and \(y\) respectively.
Then the area of the region,
\(xy = 60000\)     \(……… (i)\)
Then the cost of fencing,
\(200x + 100x + 100y + 100y\)
\(⇒300x + 200y\)
We know that,
Arithmetic mean \(≥\) Geometric mean
\(⇒\frac {300x+200y}{2}≥\sqrt {300x×200y}\)
\(⇒300x+200y≥2\sqrt {300x×200y}\)
\(⇒300x+200y≥2\sqrt {300×200 ×\ xy}\)
\(⇒300x+200y≥2\sqrt {300×200 ×\ 6000}\)       [From eq \((i)\)]
\(⇒300x+200y≥2\sqrt {3600000000}\)
\(⇒300x + 200y ≥ 2 \times 60000\)
\(⇒300x + 200y ≥ 120000\)
Thus, we can say the cost will always be greater than \(120000\).

So, the correct option is (C): \(120000\).