Question

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

Answer 1

Answer: The dimensions are length 600metres and breadth= 300 metres

Explanation:

Area of rectangle =l × b

L b =180,000

b = 180,000/L ....... equation 1

Perimeter of rectangle =2L + b

P = 2(180,000)/L + L

P= 360,000/L +L

dp/dL= -360,000/L^2 +1

L^2= 360,000

L= Squareroot of 360,000

L=600m

b= 180,000/600

b= 300m

Answer 2

The required dimensions will be "600 by 300". A complete solution is below.

According to the question,

• xy = 180000

No fencing along river so,

• $f(x) = x+2y$
• $f(x) = \frac{180000}{y+2y}$

By taking the derivative, we get

→ $f'(x) = -\frac{180000}{y^2+2} =0$

                  $\frac{180000}{y^2} = 2$

                       $y^2 = \frac{180000}{2}$

                       $y^2 = 90000$

                        $y = \sqrt{90000}$

                           $= 300$

and,

→ $x = \frac{180000}{300}$

     $= 600$

Thus the above answer is right.    

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