Question
The annual growth of a tree is 80% of its growth during the previous year.
This year, the tree is 42 m in height, and one year ago, its height was 37 m.
(a) Calculate the annual growth of the tree in the coming year.
(b) Calculate the height of the tree 6 years from now. Give your answer correct to the nearest cm.
If the tree continues to follow this pattern of growth, its height will never exceed \( k \) metres.
(c) Find the smallest possible value of \( k \).
▶️ Answer/Explanation
Detailed Solution
(a) Calculating the Annual Growth
- The growth in one year is the difference in height:
- \[ 42 – 37 = 5 \text{ m} \]
- The growth in the coming year is 80% of the previous year’s growth:
- \[ 5 \times 0.8 = 4 \text{ m} \]
(b) Finding the Height After 6 Years
Recognizing a geometric sequence where:
- Initial growth: \( 5 \) m
- Common ratio: \( r = 0.8 \)
We use the sum formula for a geometric series:
\[ S_n = a \frac{1 – r^n}{1 – r} \]
To find the total height after 6 years:
\[ 42 + \frac{4(1 – (0.8)^6)}{1 – 0.8} \]
OR
\[ 37 + \frac{5(1 – (0.8)^7)}{1 – 0.8} \]
Solving:
\[ = 56.7571… \]
Correct to the nearest cm:
\[ \mathbf{56.76} \text{ m OR } 5676 \text{ cm} \]
(c) Finding the Limiting Height \( k \)
The height will never exceed the sum of an infinite geometric series:
\[ S_{\infty} = a \frac{1}{1 – r} \]
Using the total sum formula:
\[ S_{\infty} = 37 + \frac{5}{1 – 0.8} \]
OR
\[ S_{\infty} = 42 + \frac{4}{1 – 0.8} \]
\[ k = 62 \]
Conclusion: The maximum height the tree can reach is 62 m.
……………………………Markscheme……………………………….
- (a) Annual growth: 4 m
- (b) Height after 6 years: 56.76 m (or 5676 cm)
- (c) Maximum height: 62 m