<p>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.</p>
<p>At the start of the first week of weeding, ${500}$ m${^{2}}$ of the land is covered with weed.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">In option 1, the farmer removes weed from ${10\%}$ of the area covered with weed at the end of each week.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the area covered with weed at the end of the second week.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">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.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the area covered with weed at the end of a week in the long run.</span></div><p style="text-align: right;">[1]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text">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.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that the change in the area covered with weed in the ${{{n}^{\mathrm{th}}}}$ week is given by $\left( 40-10n \right)$m${^{2}}$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, find the area covered with weed at the end of the nth week in terms of $n$.</span></div><p style="text-align: right;">[3]</p></li></ol></li></ol>

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.

In option 1, the farmer removes weed from ${10\%}$ of the area covered with weed at the end of each week.

Find the area covered with weed at the end of the second week.

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.

Find the area covered with weed at the end of a week in the long run.

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.

Show that the change in the area covered with weed in the ${{{n}^{\mathrm{th}}}}$ week is given by $\left( 40-10n \right)$m${^{2}}$.

Hence, or otherwise, find the area covered with weed at the end of the nth week in terms of $n$.

2015 DHS P2 Q4

2015 DHS P2 Q4

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