2023 MI P1 Q12

A curve \(C\) has parametric equations

\(x=t-\frac{10}{t}\), \(\qquad\) \(y=6t-{{t}^{2}}\), \(\qquad\) for \(\sqrt{10}\le t\le 6\).

1. Sketch the curve \(C\), labelling the coordinates of any points of intersection with the axes.[2]

It is given that a plot of flat land is represented by the region bounded by \(C\) and both axes, where units are measured in kilometres.

2. Find the area of the plot of land.[3]

The landowner considers using a portion of this land to rear cattle. This portion is a rectangular piece of land, \(OPRQ\), where \(O\) is the origin, \(R\) is a point on the curve, \(P\) and \(Q\) are the points on the \(x\)-axis and \(y\)-axis respectively.

3. By using differentiation, find the value of \(t\) that maximises the area of rectangle \(OPRQ\). (You need not show that your answer gives a maximum.)[3]

It is given instead that the landowner decides to divide the plot of flat land equally into two by building a straight fence through the land, parallel to the \(y\)-axis. It is assumed that the thickness of the fence is negligible.

4. Using the result from part (ii), find the equation of the fence that divides the plot of land equally into two parts.[4]