Question 27599: Amanda has 400 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). She wants to maximize the area of her patio (area of a rectangle is length times width). What should the dimensions of the patio be?
Answer by mbarugel(146)   (Show Source):
You can put this solution on YOUR website! Let's call L to the length of the rectangle and W to its width.
We know that the perimiter of the rectangle must be 400, since that is all the lumber Amanda has available. Therefore, we have the equation:
Let's isolate L:
Now, we know that the formula for the area of a rectangle is W*L. Therefore, using the equation we've just found, the area of this rectangle will be:
Distributing:
So the area, as a function of the width, is a parabola. Now, when the quadratic coefficient of a parabola is negative (as in this case, in which it's -1), the maximum value of the parabola is achieved at its vertex. Finally, recall that the formula for the vertex of a parabola is:
where b is the linear coefficient (in this case +200) and a is the quadratic coefficient (in this case, -1). Therefore, the vertex of this parabola is at:
The width of the rectangle must be 100. Plugging this into , we get that the length must also be 100. Therefore, the area is maximized when the patio is a square of length 100. Total Area is 100*100 = 10,000.
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