Amanda has 400 feet of lumber to frame a rectangular patio?

Question

(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, and show how the maximum area of the patio is calculated from the algebraic equation. Use the vertex form to find the maximum area.

can u try to show the work?

Answer 1

P = 2(L + W) = 400 ft.
A = LW
L + W = 200
L = 200 - W
A = (200 - W)W
A = 200W - W^2
A = -(W^2 - 200W + 10,000 - 10,000)
A = -(W - 100)^2 + 10,000
A - 10,000 = -(W - 100)^2
This is a parabola opening downward with a vertex at (100, 10,000), which will be the width and the maximum area.
W = 100 ft. so L must also be 100 ft., giving you a square of 10,000 ft^2 area.

Answer 2

Since you need to show your work, you can't simply say that it's a square...it is, but you need to prove it. (Yeah, it sucks, I never liked these either) The way to do this is to write out the equation you know,

2l+2w=400

That is; 2*length (l) + 2 * width (w) gives us the perimeter, 400. Now we re-write that to solve for either l or w...I'll solve for l.

2l=400-2w

l=(400-2w)/2

l=200-w

Now we take that and substitute it into the basic area formula for a rectangle (l*w=a) and it'll be:

(200-w)*w=area

multiply everything within the parenthesis to get rid of them:

200w-w^2=area, then we use the vertex formula (since thats what the question said to do)...the vertex formula gives us the x-coordinate of the vertex, which will be the maximum for this parabola:

Vertex form.= -b/2a

in our equation, b=200 and a= -1, so we'll have:

-200/-2=100

Therefore, the x-coordinate of the vertex is 100. Pop that right back into the perimeter equation we have:

2(100)+2y=400

200+2y=400

2y=200

y=100

So the dimensions are: length = 100, width=100, a square...

Answer 3

Sum of the length of all 4 sides = 400 ft

That is 2W + 2L = 400 ft
2W = 400 - 2L
W = 200 - L

Area A = L * W
A = L * (200 - L)
= 200 L - L^2

When the area is maximum dA/dL =0
ie. 400 - 4 L = 0
4 L = 400
L = 100

W = 200 - 100
=100

A = 100 * 100 = 10,000 sq. ft

Answer 4

It is a general principle that if you are trying to maximize the area of a rectangle with a given perimeter, it must be a square. I have never heard of the "vertex form"; the proof of the principle is a simple problem in differential calculus.

Answer 5

If you can't do basic geometry, then maybe you need to get off the computer and study.

Answer 6

What has Amanda ever done for me?