Kindly prepare a Figure as described below :
Consider a Rectangle #ABCD# inscribed in a Semi-circle with
Diameter #PQ# and the Centre #O,# such that, side #AB#
lies on #PQ# and the the vertices #C and D# are on the semi-circle.
The collinear points #O,A,B,P,Q# are in the order #P,A,O,B,Q#
from left-to-right.
Let #/_COB=theta,# and, #r# be the radius. #:. OC=r=18.#
In the right-#DeltaCOB, OB=OCcostheta, CB=OCsintheta.#
#:. OB=rcostheta, CB=rsintheta. rArr AB=2OB=2rcostheta.#
#:."The Area of the rectangle ABCD="ABxxBC,#
#=2rcostheta*rsintheta=r^2(2sinthetacostheta)=r^2sin2theta.#
Since, #r# is constant, we conclude, from this, that the Area will be
maximum, when so is #sin2theta.#
But the maximum value of #sin2theta# is #1,# corresponding to
#2theta=90, or, theta=45.#
Therefore, the Maximuum Area of the Rectangle in question is
#r^2, i.e., 18^2=324" sq. units."#
Enjoy Maths.!
