A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the `x y` plane bounded by the lines `y=0,y=3x ,a n dy=30-2xdot` If `M` is the largest area of such a rectangle, then the value of `(2M)/(27)` is_______

Answer

67.5

Explanation

Area A=`(X_(2)-x_(1))y`
`y=3x_(1) and y=30-20x_(2)`
`therefore A(y)=(30-y)/(2)-(y)/(3)y`
or `6A(y)=(90-3y-2y)(y=90y-5y^(2))`
So 6A(y)=90-10y=0
or `y=9, A''(10)=-10lt0`
`therefore x_(1)=3,x_(2)=21/2`
`therefore A_(max)=(21/2-3)9=(15xx9)/(2)=(135)/(2)`