2015 DHS P2 Q4

A farmer owns a parcel of land that is partially covered with weed. The area that is covered with weed increases by \({80}\)m\({^{2}}\) each week. The farmer decides to start weeding.

At the start of the first week of weeding, \({500}\) m\({^{2}}\) of the land is covered with weed.

1. In option 1, the farmer removes weed from \({10\%}\) of the area covered with weed at the end of each week.
   1. Find the area covered with weed at the end of the second week.[2]
   2. Show that the area covered with weed at the end of the nth week is given by \({\left( {{0.9}^{n}}\left( 500 \right)+k\left( 1-{{0.9}^{n}} \right) \right)}\)m\({^{2}}\) where \({k}\) is a constant to be determined.[3]
   3. Find the area covered with weed at the end of a week in the long run.[1]
2. In option \({2}\), the farmer removes weed from an area of \({50}\) m\({^{2}}\) at the end of the first week. He removes weed from an additional area of \({10}\) m\({^{2}}\) at the end of each subsequent week. Thus he removes weed from an area of \({60}\) m\({^{2}}\) at the end of the second week, and \({70}\) m\({^{2}}\) at the end of the third week, and so on.
   1. Show that the change in the area covered with weed in the \({{{n}^{\mathrm{th}}}}\) week is given by \(\left( 40-10n \right)\)m\({^{2}}\).[1]
   2. Hence, or otherwise, find the area covered with weed at the end of the nth week in terms of \(n\).[3]