Question 1023080: The area of a rectangle with a perimeter of 120 units can be described by y = x (60 - x), where x represents the width of the rectangle and y represents the area. A farmer has 120 yards of fencing available and wishes to enclose a rectangular area. To the nearest five years, what width gives the largest enclosed area?
Found 2 solutions by josmiceli, solver91311:
Answer by josmiceli(19441)   (Show Source):
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
To the nearest 5 years? What are you talking about?
Let a rectangle have a perimeter  , then if the length is  and the width is  , we can say that  , and conclude that 
Since we know that the area of the rectangle is length times width, we can write an expression for area as a function of width by saying:
\ =\ -w^2\ +\ \frac{P}{2}w)
Since the vertex of the parabola \ =\ ax^2\ +\ bx\ +\ c) is at the point ) where:
 = f\left(\frac{-b}{2a}\right))
So, for the area function we derived above, the maximum value of the function is where:
Which is to say, you get the maximum area when the width is exactly one-fourth of the perimeter. The only way for the width to be exactly one-fourth of the perimeter is for the rectangle to have four equal measure sides, i.e. a square.
John
My calculator said it, I believe it, that settles it
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