# What should the dimensions of the larger pasture be if he wants to use the least amount of fencing in the following statementA farmer wants to divide his 2700 km^2 of land into two separate...

What should the dimensions of the larger pasture be if he wants to use the least amount of fencing in the following statement

A farmer wants to divide his 2700 km^2 of land into two separate pastures for his new horses by building a fence that will enclose both pastures. He wants to pasture to be twice the size as the other. What should the dimensions of the larger pasture be if he wants to use the least amount of fencing?

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### 2 Answers

I am going to suppose that the land is rectangular, in this case the width for both pastures will be the same. Let x be the length of the smaller pasture, then 2x should be the length os the second one. Area of first is wx area of second is 2wx.

So dimension of the original land will be w by 3x. Since the area is 2700, then `w=2700/(3x)`

Amount of fencing needed is then given by the following function

`f(x)=3w+2(3x)=3*(2700/(3x))+6x=>`

` ``f(x)=2700/x+6x`

To find the min point we need to find the first derivative, set it equal to zero and solve.

`f'(x)=-2700/(x^2)+6=>`

`f'(x)=0=>-2700/(x^2)=-6=>x^2=2700/6=450`

Since we are working with dimensions, we are only interested in the positive roots, so `x=sqrt(450)=15sqrt(2)`

**So the dimension of the larger pasture should be** `30sqrt(2)~~21.21Km`

**by** `127.30 Km`

i had the same question in my math study guide however options for the answer is

a)30km*30km

b)60km*60km

c)60km*90km

d)60km*30km