SOLUTION: Amanda has 400 feet of lumber to frame a rectangular patio. She wants to maximize the area of her patio. What should the dimensions of the patio be, and show how the maximum area

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Question 56158: Amanda has 400 feet of lumber to frame a rectangular patio. She wants to maximize the area of her patio. What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation. Use the vertex form to find the maximum area.
Answer:

Show work in this space.
Answer by funmath(2933) (Show Source):
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Amanda has 400 feet of lumber to frame a rectangular patio. She wants to maximize the area of her patio. What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation. Use the vertex form to find the maximum area.
Answer: A=10,000 square ft

Show work in this space.
Let length=x
Let width=y
Perimeter=400
2(length)+2(width)=Perimeter
2x+2y=400
2y=400-2x
2y/2=400/2-2x/2
y=200-x
A=length*width
A=xy but y=200-x so
A=x(200-x)
Vertex form is , where (h,k) is the vertex.

We need to create a perfect square. We do that by adding (-200/2)^2=(100)^2=10000 to the inside of the parenthesis. Because of the - infront of the parenthesis we are really taking 10000 away, so we have to add the same thing on the outside of the parenthesis so that we are in reality only adding 0.
A=-(x^2-200x+____)+____
Now we have to factor the perfect square we created. . b^2=10000, so

The vertex is (100, 10000), that means that the maximum x (length) is 100 ft, and the maximum A (area) is 10000 ft^2
I hope this is a little more clear.
Happy Calculating!!!