Question regarding dimensions of a rectangle?

Question

Among all rectangles having a perimeter of 25m, find the dimensions of the one with the largest area.

How do I begin to solve this?

Answer 1

a square has the largest area of all rectangles. divide 25 into 4 equal lengths and you will have the rectangle with the largest area
EX
25/4=6.25
6.25x6.25=39.0625

Answer 2

The area of a rectangle is length times width (L*W=A). We're given that the perimeter is 25, meaning that 2L+2W=25, or that L+W=12.5, which can be rearranged to W=12.5-L. Going back to the area formula, we can substitute 12.5-L for W to get

L*(12.5-L)=A
12.5L-L^2=A

If you know calculus, you can just take the derivative of the left side and set it equal to 0 to get...

12.5-2L=0
12.5=2L
6.25=L

with the width being 12.5-6.25= 6.25 as well.

If you don't know calculus you can try making a table of different lengths and the areas that you get by calculating them out, then realizing that the closer that L is to W (the closer the rectangle is to a square) the larger the area, and then surmise that the answer is when the rectangle is a square, L=W, and the each would be a fourth of 25m, or 6.25m.

Answer 3

It has to be a square, so divide your perimeter by 4 = 6.25m would be the length of each side. To prove that it has to be a square, suppose you let s = the square's side length, and you lengthen one side by t, and shorten the second side by t (to keep the same perimeter). Then the area would be equal to l x w or (s+t) x (s-t). Expand that, and you get s^2 - t^2. No matter what you pick for t, the area is always less than s^2, the area of the square.

Answer 4

The rectangle with the largest area will be a square.

That should tell you all you need to know.

Answer 5

The answer is 6.25x6.25 (square)

Solution:
Let x and y be the sides of the rect. So 2x+2y=25 or x+y=12.5 or y=12.5-x. The area of the rect will be a(x)=x*y=x*(12.5-x)=-x^2+12.5*x. To find the maximum of that function you must find it's critical values - where a'(x) gets zero and where a(x) is not continual - which are only where a'(x) gets zero. So:

a'(x)=0
(-x^2+12.5*x)'=0
-2*x+12.5=0
2*x=12.5
x=6.25

Because a(x) is a polynomial of 2nd degree and because the coefficient of x^2 is negative 6.25 will be the maximum of a(x). Because it is in the region (0,12.5) of a(x) this is the answer.

Answer 6

area = xy (length * width), where 2x+2y=25.

So area = xy = x(25-2x)/2. Differentiate to maximize.

Actually, you don't need to differentiate. By symmetry, the area in terms of y is
y = y(25-2x)/2, the same function.

So whatever value of x maximizes the area, the same value for y will do it as well. Therefore we are optimal when x=y... a square.

Answer 7

Make a list of 2 numbers that add up to 25.
Like:
24 1
23 2
22 3

then multiply each pair of numbers.
Like:
24
46
66

then list the numbers that have the largest area!
Like:
12 13

Answer 8

p= 2l+2w =25 --> l+w =12.5
A=lw = l(12.5-l) = 12.5l-l^2
dA/dl = 12.5-2l
Set this to 0 to find max
2l = 12.5
l = 6.25 = length
w = 12.5 - 6.25 = 6.25 = width.
Thus max area uccurs when rectangle is a square.