Question 606060
Let L = length and W = width


Draw out a picture to get

<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/barn_garden.png">

From the picture above, the fence of the garden is represented by the blue lines. Notice that there isn't a 4th blue line at the top because the barn wall is used here.


Since we have 3 sides (that are W, L and W units long), the perimeter of the garden (excluding the barn wall) is...


W+L+W = 2W+L


So the length of fencing needed is 2W + L feet. But we're given that we have 100 ft of fencing. So if we use every bit of that fencing, we know that


2W + L = 100



Solve for L:



2W + L = 100


2W + L -2W = 100 - 2W


L = <font color="red">100 - 2W</font>


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Now onto the area of this garden. 

The area of any rectangle is just the result of multiplying the length by the width. 


So A = LW


But we know that the length is L = <font color="red">100 - 2W</font>, so replace L with <font color="red">100 - 2W</font>



A = LW


A = (<font color="red">100 - 2W</font>)W


A = W(100 - 2W)


Now simplify



A = W(100 - 2W)


A = 100W - 2W^2


A = -2W^2 + 100W


Hopefully you're with me. If not, go back and make sure you understand where we're at and how we got here.


We now have the equation A = -2W^2 + 100W


This is an area function that depends on the width W. 
This function will plot out a parabola. 
It turns out that this parabola will open downwards, which means that it peaks somewhere.



This peak is the vertex. To find the vertex, we need to find the x coordinate of the vertex. It can be found by the formula


x = -b/(2a)


In the case of A = -2W^2 + 100W, a = -2 and b = 100, so the x coordinate of the vertex is 


x = -100/(2(-2))


x = -100/(-4)


x = 25


So from here, just replace 'x' with W to get W = 25


This means that at this coordinate, the area function will max out. 


Now just plug this value into the area function to actually compute the max area


A = -2W^2 + 100W


A = -2(25)^2 + 100(25)


A = -2(625) + 100(25)


A = -1250 + 2500


A = 1250


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<font color="red">Answer:</font>


The maximum area the farmer can enclose with 100ft of fence is _<u><font color="red">1250</font></u>_ sq. ft


The dimensions of the garden to give this area is 50ft by _<u><font color="red">25</font></u>_ ft


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