fence optimiztion question

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fence optimiztion question

Postby Integr88 » Thu 31 March, 2011 16:04

A former wishes to built a fence along a river. He has 240meters of fence. What are the dimensions which he can build a fence with maximum area?
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Re: fence optimiztion question

Postby barnamah » Thu 31 March, 2011 16:17

Because he is building fence along the river, we have an area that is fenced on three side only.

The perimeter is x+x+y=240 or
`2x+y=240` eq (1)

We also have area:
`A=x*y` eq (2)

Now in eq(1) solve for y by subtracting 2x from both side:
`y=240-2x` eq (3)

Now using eq(2) we sub in the y value from eq(3)
`A=x*y`
`A=x(240-2x)` and FOIL we have `A=240x-2x^2`. Now take the derivatives:

`A'=240-4x` eq(4)
Set the this to 0 and we have: `0=240-4x` and solve for x. Subtract 240 from both side and we have:
`-240=-4x`

divide both side by -4 and we have:
`-240/-4=(-4x)/4`. 4 on the right side cancels out and have: `-240/-4=x` which is x=60

So we have x=6.

Find y by suing x=6 in eq(3) which is `y=240-2x` and we have:
`y=240-2(60)=240-120=120`
so y=120

the length of the fence should be 120 and width 60 which gives: 120*60=7200

`:.` final answer: dimensions should be 120 by 60
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