2024 VJC P1 Q11

Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass various activities related to the identification, classification, biology, and ecology of trees. A group of dendrologists are studying the growth of \(2\) species of trees, codenamed Tree Vee and Tree Jay.

In the 1st year, the height of Tree Vee and Tree Jay are both \(H\) cm. In the 2nd year, Tree Vee’s height increases by \(s\) cm and subsequently, the increase in height every year is \(10\%\) less than the previous year’s increase.

1. Show that the height of Tree Vee in the 4th year is given by \(\left( H+2.71s \right)\) cm.[1]
2. Show that the height of Tree Vee in the \({{n}^{\mathrm{th}}}\) year is given by \(\left[ H+10s\left( 1-{{0.9}^{n-1}} \right) \right]\) cm.[3]
3. Hence, write down in terms of \(H\) and \(s\), the theoretical maximum height (in cm) of Tree Vee.[1]
4. In the 2nd year, Tree Jay’s height increases by \(t\) cm and subsequently, the increase in height every year is \(0.5\) cm less than the previous year’s increase. Show that the height of Tree Jay in the \({{10}^{\mathrm{th}}}\) year is given by \(\left( H-18+9t \right)\) cm.[2]

It is now given that \(t=20\).

5. After the \({{10}^{\mathrm{th}}}\) year, Tree Jay’s height increases at a constant rate of \(7\) cm per year. Express Tree Jay’s height (in cm) in the \({{n}^{\mathrm{th}}}\) year (where \(n\ge 11\)) in terms of \(H\) and \(n\).[2]
6. It is further given that \(s=30\), and the 1st year is the year 2024. Find the years in which the heights of Tree Vee and Tree Jay are within 7 cm of each other, after 2034.[3]