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. 

Answer 1

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