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

Answer 1

#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#

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Answer 2

Cameron Alexander

To find the largest possible area for a rectangle inscribed in a circle of radius 4, the rectangle's length should be twice the circle's radius, and its width should be equal to the circle's radius. Thus, the largest possible area ( A ) is given by:

[ A = \text{{length}} \times \text{{width}} = 2 \times 4 \times 4 = 32 , \text{{square units}}. ]

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Answer from HIX Tutor

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