Question 524005: find the dimensions of a rectangle area, which is to have perimeter of 100''. and which has maximum area Found 2 solutions by nerdybill, solver91311:Answer by nerdybill(7384) (Show Source):
You can put this solution on YOUR website! find the dimensions of a rectangle area, which is to have perimeter of 100''. and which has maximum area
.
Let x = width
and y = length
then, from perimeter
2(x+y) = 100
x+y = 50 (equation 1)
.
and, from area
area = xy (equation 2)
.
Solve equation 1 for y:
x+y = 50
y = 50-x
.
Substitute above into equation 2:
area = xy
area = x(50-x)
area = 50x-x^2
MAX is when:
x = -b/(2a)
x = -50/(2(-1))
x = -50/(-2)
x = 25 feet
.
substitute above into equation 1 to find y:
x+y = 50
25+y = 50
y = 25 feet
.
Max area is attained by a square measuring
25 ft by 25 ft
The dimensions of a rectangle with maximum area with a given perimeter is a square with sides that measure the perimeter divided by 4.
The perimeter of a rectangle is given by:
The area of a rectangle is given by:
Substituting from the perimeter equation:
This function graphs to a parabola opening downward meaning that the vertex is a maximum. The maximum value of the function, hence the maximum area, is where the value of the first derivative is equal to zero:
Set equal to zero:
Hence, the maximum area rectangle for a given perimeter is a square with sides of length one-fourth of the perimeter.
John
My calculator said it, I believe it, that settles it
