How do you find the largest possible area for a rectangle inscribed in a circle of radius 4?

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Answer 1 · 2016-09-18T10:57:34

#32# units of area

An inscribed rectangle has diagonals of length #2r# and has sides #(a,b)# measuring #0< a < 2r#, #b = sqrt((2r)^2-a^2)# so the rectangle area is

#A = a sqrt((2r)^2-a^2)# for #0< a < 2r#.

Its maximum occurs at #a_0# such that

#((dA)/(da))_(a_0) = 0# or

#(2 (a_0^2 - 2 r^2))/sqrt[4 r^2-a_0^2] = 0# giving

#a_0 = sqrt(2)r# and at this value

#A_0 = 2r^2=2xx4^2= 32#