\(OABC\) is a square of side \(\quantity{1}{ft.}\). A point \(X\) in \(AB\) and a point \(Y\) in \(BC\) are such that \(AX=\quantity{x}{ft.}\), \(BY=\quantity{kx}{ft.}\) and \(BY\) is longer than \(AX\). For a given value of \(k\), show that the minimum area of the triangle \(OXY\) is \[\quantity{\frac{4k-1}{8k}}{sq.ft.}\]

Could we draw a picture of the problem?

Can we write the area of the triangle \(OXY\) as a function of \(x\)?

The bar labelled ‘Area’ has a height that measures the area of triangle \(OXY\).

The bar labelled ‘Bound’ shows the value of \(\dfrac{4k-1}{8k}\).

For a given \(k\), how does the area vary with \(x\)?