Mike, an amateur astronomer who lives in the South of England, wants to know how the number of hours of darkness changes through the year - AQA - A-Level Maths Pure - Question 8 - 2020 - Paper 1

We need to solve the inequality: $H > 14$ Substituting in the model: $3.87 \text{sin}\left( \frac{2\pi(t + 101.75)}{365} \right) + 11.7 > 14$. This simplifies to: $\text{sin}\left( \frac{2\pi(t + 101.75)}{365} \right) > \frac{14 - 11.7}{3.87} = 0.59$. We now find the corresponding t values. The sine function exceeds 0.59 approximately between: $t_1 \approx 300.22,\quad t_2 \approx 408.77$ Thus, the interval of t where these conditions hold leads to: $t_2 - t_1 = 408.77 - 300.22 \approx 108.55\text{ days.}$

Sofia's refinement, which increases the 3.87 value while keeping the other parameters constant, would increase the amplitude of the graph, thereby increasing the maximum fluctuation of hours of darkness. However, the original data suggests that this may not be necessary, as the graph of her data appears to have a lower amplitude compared to Mike's model. Hence, Sofia's refinement may not be appropriate, as it does not accurately align with the observed data, potentially leading to misleading conclusions.