A tree increases annually by $\frac{1}{8}$ of its height. By how much will it increase after $2 \frac{1}{2}$ years if it stands today 8 m high?

### Concept & Formula When an object grows at a constant fractional rate relative to its current size, it follows compound growth. For fractional years (like $2 \frac{1}{2}$ years), we apply the full compound growth formula for the whole years, and then apply simple growth for the remaining fractional year on the final compounded amount. Formula for fractional years: $$ A = P \left(1 + R\right)^n \left(1 + \text{fraction} \times R\right) $$ Where $R$ is the fractional rate of increase. ### Step-by-Step Solution * **Given:** Initial height ($P$) = $8\text{ m}$. Annual growth rate ($R$) = $\frac{1}{8}$. Time = $2 \frac{1}{2}$ years. * **Calculate height after 2 full years:** $$ H_2 = 8 \times \left(1 + \frac{1}{8}\right)^2 $$ $$ H_2 = 8 \times \left(\frac{9}{8}\right)^2 $$ $$ H_2 = 8 \times \frac{81}{64} = \frac{81}{8} = 10.125\text{ m} $$ * **Calculate growth for the final $\frac{1}{2}$ year:** The rate for half a year is $\frac{1}{2} \times \frac{1}{8} = \frac{1}{16}$. $$ H_{final} = 10.125 \times \left(1 + \frac{1}{16}\right) $$ $$ H_{final} = 10.125 \times \frac{17}{16} $$ $$ H_{final} \approx 10.125 \times 1.0625 = 10.7578\text{ m} $$ * Rounding to two decimal places to match the options gives $10.75\text{ m}$. ### Exam Strategy & Shortcut **Step-by-Step Manual Addition:** For small numbers and short timeframes, manual sequential addition is often faster and less prone to formula errors. * **Start:** $8\text{ m}$ * **End of Year 1:** Grows by $\frac{1}{8}$ of $8$ ($1\text{ m}$). New height = $9\text{ m}$. * **End of Year 2:** Grows by $\frac{1}{8}$ of $9$ ($1.125\text{ m}$). New height = $10.125\text{ m}$. * **Half of Year 3:** Grows by $\frac{1}{16}$ of $10.125$ (roughly $0.63\text{ m}$). * **Final:** $10.125 + 0.63 = 10.755\text{ m}$. The closest option is 10.75 m. ### Common Pitfall *Ambiguous Phrasing Trap:* The question asks "By how much will it increase," which technically means finding the difference (Final Height - Initial Height). However, looking at the options provided ($10.75$, $11.85$, etc.), they clearly represent the *final height* itself, not the increase. This is a common phrasing error in competitive exam papers. Always cross-reference your calculated delta with the given options to infer the examiner's true intent. ### Final Answer **Therefore, the correct answer is 10.75 m.**