Question 199484: Problems like these always make sure I come back to this website for help =( ; Find the dimension of a rectangle “a” with the greatest area whose perimeter is 30 ft.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The perimeter of a rectangle is given by the formula:
Where  is the measure of the length of the rectangle and  is the measure of the width.
Solving for :
The area of a rectangle is given by the formula:
Substituting the expression for derived earlier, you can write a function for the area of the rectangle in terms of the width where the perimeter is a constant:
 = \left(\frac{P - 2w}{2}\right)w = - w^2 + \frac{1}{2}Pw)
This is a quadratic function whose graph is a parabola. Since the lead coefficient is <0, the parabola opens downward and the vertex of the parabola represents a maximum. Since the independent variable is the width and the value of the function is the area, the coordinates of the vertex will tell us the width in terms of the perimeter that gives us the maximum area, and the value of that maximum area, again in terms of the perimeter.
A parabola represented by  = ax^2 + bx + c) has a vertex at the point:
\right))
For the area function derived above,  and  , so:
Hence, the maximum area is obtained when  , which means that  and therefore  also. Therefore, the maximum area for a given perimeter is obtained when the rectangle is actually a square with side measure one-fourth of the perimeter.
John
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