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Question
A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
Solution
Breadth = x, length = `2sqrt("r"^2-"x"^2)`
A = `2"x"sqrt("r"^2-"x"^2)`
`"dA"/"dx" = 2 sqrt("r"^2-"x"^2) + (2"x")/(2sqrt("r"^2-"x"^2) )(-2"x")`
`("d"^2"A")/("dx"^2) = 2/(2sqrt("r"^2-"x"^2))( -2"x") - (4"x")/sqrt("r"^2-"x"^2) +(2"x"^2)/(2("r"^2-"x"^2)^(3/2)) (-2"x")<0`
Hence, area is maximum
Point of maxima is given by : `"dA"/"dx" = 0`
⇒ `(2("r"^2 -"x"^2 -"x"^2))/sqrt("r"^2-"x"^2) = 0`
⇒ `"x"="r"/sqrt2`
∴ Breadth =`"r"/sqrt2,` length =`sqrt2"r"`
Maximum area = r2
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