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

Set up the inequality: $H = 3.87\sin\left(\frac{2\pi(t + 101.75)}{365}\right) + 11.7 > 14$ This simplifies to: $3.87\sin\left(\frac{2\pi(t + 101.75)}{365}\right) > 2.3$ Calculating: $\sin\left(\frac{2\pi(t + 101.75)}{365}\right) > \frac{2.3}{3.87} \approx 0.593$

Finding $t$ values using the sine function gives two critical points:

• $t_1 \approx 300.22$
• $t_2 \approx 408.77$ Then, calculating the length of consecutive days gives: $408 - 300 = 108.$ Thus, the maximum number of consecutive days is 108.

Sofia's refinement, which increases the amplitude of the sine function, would raise the maximum value of darkness hours predicted by the model. However, the data from her own observations indicates a different seasonal characteristic.

Sofia's graph suggests that the fluctuation in hours of darkness is not as pronounced as Mike's model might indicate. Thus, while increasing the amplitude could provide a theoretical maximum, it does not reflect the actual variability observed. Therefore, her refinement is not appropriate, as it misrepresents the data.