let

l = the length of the rectangle, l>0

w = the width of the rectangle, w>0

2(l + w) = 40 divide bot sides by 2

l + w = 20

l = 20 - w

the area of the rectangle A = l*w

plug l = 20 - w into A = l*w

A = (20 - w)*w

A = - w^2 - 20w

we need to find w so that the area A = -w^2 - 20w is maximum, that is find the vertex of the parabolic function A = -w^2 - 20w

w = -b/2a, where a = -1, b = -20

w = -(-20)/(-2(-1))

w = 10 m

l = 20 - 10

l = 10 m

the dimensions of the rectangle with the greatest area are: the length if equal to the width that is equal to 10 meters.

this rectangle is a square.