Question 606060: Word problems STRESS me out!!! Please help!
A farmer decides to enclose a rectangular garden using the side of the barn as one side of the rectangle. What is the maximum area the farmer can enclose with 100ft of fence? What should the dimensions of the garden be to give this area?
THe maximum area the farmer can enclose with 100ft of fence is ___ sq. ft?
The dimensions of the garden to give this area is 50ft by ___ ft?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let L = length and W = width
Draw out a picture to get
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 = 100 - 2W
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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 = 100 - 2W, so replace L with 100 - 2W
A = LW
A = (100 - 2W)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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Answer:
The maximum area the farmer can enclose with 100ft of fence is _1250_ sq. ft
The dimensions of the garden to give this area is 50ft by _25_ ft
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Jim
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