SOLUTION: A rectangle has a perimeter of 100 cm. Find the greatest possible area for the rectangle.
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Question 267316: A rectangle has a perimeter of 100 cm. Find the greatest possible area for the rectangle.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
The perimeter of a rectangle is given by:
So the length in terms of the perimeter and the width is:
The area of a rectangle is given by:
Substituting the previously derived expression for the length we can generate a function for Area in terms of the width:
\ =\ w\frac{P\ -\ 2w}{2})
Which is to say:
\ =\ \frac{Pw\ -\ 2w^2}{2})
And this can be recognized to be a quadratic function. Since the lead coefficient when the function is in standard form is negative, the graph of this function is a parabola opening downward. The vertex is therefore a maximum.
Using the formula for the 
-coordinate of the vertex of the parabola, will be able to determine that 
.
Hence, the area is maximum when the width of the rectangle is the perimeter divided by 4. If the width of the rectangle is one fourth of the perimeter, the length must also be one-fourth of the perimeter. Therefore for any given perimeter, the maximum area rectangle that can be formed is a square with side measure one-fourth of the perimeter.
John
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