If a rectangle has a perimeter of 36 cm, find the dimensions that make the area of the rectangle maximum without using calculus.
The dimensions of a rectangle with perimeter 36 cm have to be determined that maximize the area without using calculus.
Let the length of the rectangle be L and the width be W. The perimeter of a rectangle is 2(L + W) = 36
=> W = 18 - L
The area of the rectangle is A = L*W = L*(18 - L)
=> 18L - L^2
The graph of A = 18L - L^2 has been plotted below with area on the y-axis and length L on the x-axis
As can be seen the graph peaks at A = 81. At A = 81, L = 9
The length of the required rectangle has a length equal to 9 and the width is 18 - 9 = 9.
The required rectangle is a square with sides equal to 9 cm and the maximum area is 81 cm^2.