A rectangle has perimeter equal to 28, what is the maximum possible area of this rectangle.
x y = A
y = A/x
2 x + 2 y = 28
so
x + y = 14
x + A/x = 14
x^2 + A = 14 x
x^2 - 14 x = -A
x^2 - 14 x + 49 = -A + 49
(x-7)^2 = -(A-49)
parabola with x = 7 and A = 49
in other words 7 by 7 square
Any 2 numbers that multiply to 28 will work.
I choose 2 and 14. these are the combined lengths of the hight and length, respectively so hight is 1 and length is 7. Multiply them, your answer is the area, in units, squared, (like square feet...or SqFt. or Ft^2)
But Ken, it asks for the Maximum.
You altered something there, the total perimeter is 28, how did you get one with 49?
The AREA is 49, the PERIMETER is 28
You are to maximize the AREA
Maximimum area = 7 * 7 = 49
perimeter required is 28, or 4 * 7
2 by 14 by the way gives area of 28
BUT perimeter of
2*2 + 2*14 = 4 + 28 = 32
which simply does not work
Ah, My misread :P
Oh!, worse than a misread I stand corrected, sorry!
To find the maximum possible area of a rectangle given its perimeter, we need to use the principle that a square has the maximum area among all rectangles with the same perimeter.
Let's consider a square with side length 's'. The perimeter of this square is given by 4s, since all sides of a square are equal.
In our case, the perimeter of the rectangle is given as 28. So we can set up the equation:
4s = 28
Dividing both sides of the equation by 4, we get:
s = 7
Therefore, the side length of the square is 7.
Now, since we're looking for a rectangle, we can take this square and make it into a rectangle by elongating one of its sides. In this case, we double the length of one side, giving us a rectangle with dimensions 7 x 14.
To calculate the maximum possible area of this rectangle, we multiply the length and width:
Area = length x width = 7 x 14 = 98.
So the maximum possible area of the rectangle with a perimeter of 28 is 98 square units.