What dimensions should the garden be to maximize the area while keeping the same perimeter?

Justin: What dimensions should the garden be to maximize the area while keeping the same perimeter?
The area of the garden is 160 ft² and the perimeter is 52 ft.

How do I do this? I already know the dimensions of the garden are 10 × 16 ft but what does it mean to “maximize the area while keeping the same perimeter”? And how would you solve this?

Answers and Views:

Answer by Julia
Does it have to be a square or rectangle? If so, then the area is maximized when it is a square, so the sides would be 13 ft each and the area would be 169 ft^2.

If you can curve the edges, then the area will be maximized when it is a circle:
perimeter = 52 = 2(pi)(r), so r would be 8.28. Then, the area would be (pi)(r)^2 = (3.14)(8.28)^2 = 215 ft^2.

Answer by bpmonkey
Assuming that the garden is a quadrilateral, the question is asking what two dimensions, length and width, multiplied together would give you the greatest area.

The easiest way to do this is to simply divide the perimeter given by two. This would give you 26 in this case. Then keep multiplying two numbers that add up to 26 until you have determined which pair gives you the highest product. In this case, I believe the answer would be dimensions of 13 x 13, meaning the garden would have to be a square. Make sense?

I’m new to this whole answering questions thing so I hope this helps. haha

Answer by Josh
perimeter is given by the function
p=2w+2l
52=2w+2l

area is given by the function
a=lw
you want max area for the given perimeter
52=2w+2l
(52-2w)/2=l

maxarea=w(52-2w)/2
maxarea=(52w-2w^2)/2
maxarea=26w-w^2
if you are doing algebra, plug that into your graphing calculator and use the max feature. other wise take the derivative. maxarea becomes 0 when you take the derviative
0=26-2w
-26=-2w
w=13
52=2(13)+2l
26=2l
13=l
so the lenght and width should both be 13feet, giving an area of 169 ft^2