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lyudmila [28]
3 years ago
5

A farmer wants to fence a rectangular pasture using a river as one side. The pasture must contain 180,000 square meters of space

for the herd to frolic. What is the least amount of fencing required to enclose such a pasture?
Mathematics
1 answer:
Paul [167]3 years ago
3 0

Answer:

1200 m

Step-by-step explanation:

We are given that

Area of pasture,A=180000 square meter

We have to find the least amount of fencing required to enclose such a pasture.

Let length of pasture,=x

Width of pasture,b=y

Area of pasture=l\times b=xy

xy=180000

y=\frac{180000}{x}

Fencing required=2x+y

P=2x+\frac{180000}{x}

Differentiate w.r.t x

\frac{dP}{dx}=2-\frac{180000}{x^2}

\frac{dP}{dx}=0

2-\frac{180000}{x^2}=0

\frac{180000}{x^2}=2

x^2=\frac{180000}{2}=90000

x=\sqrt{900000}=300m

It is always positive because length cannot be negative.

y=\frac{180000}{300}=600m

Again differentiate w.r.t x

\frac{d^2P}{dx^2}=\frac{360000}{x^3}

Substitute x=300

\frac{d^2P}{dx^2}=\frac{360000}{27000000}=0.013>0

The fence is minimum at x=300

Therefore, fencing required  to enclose such a pasture

P=2x+y=2(300)+600=1200 m

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