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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
Question 8
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. On various days between F... show full transcript
Worked Solution & Example Answer: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
Step 1
Find the minimum number of hours of darkness predicted by Mike's model.
Answer
To find the minimum hours of darkness, we start by determining the minimum value of the sine function, which occurs when
[ \text{sin}\left(\frac{2\pi(t + 101.75)}{365}\right) = -1 ]
Substituting this into Mike's model gives us:
[ H = 3.87 \times (-1) + 11.7 = -3.87 + 11.7 = 7.83 \text{ hours} ]
This corresponds to: 7 hours and 50 minutes.
Step 2
Find the maximum number of consecutive days where the number of hours of darkness predicted by Mike's model exceeds 14.
Answer
To find when the hours of darkness exceeds 14, set up the inequality: [ 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{2.3}{3.87} \approx 0.593 ] The solutions for will occur at specific intervals as the sine function oscillates between -1 and 1, leading to: Two approximate values of calculations around: [ t \approx 300.22 \text{ and } 408.77 ] Thus, the consecutive days can be calculated as: [ t_{end} - t_{start} = 408 - 300 = 108 \text{ days} ]
Step 3
Explain whether Sofia's refinement is appropriate.
Answer
Sofia's increase of the 3.87 value in the model aims to increase the amplitude of the sine wave, thus potentially increasing the range of hours of darkness. However, since Sofia's data suggests a lower amplitude, this adjustment appears inappropriate. Her model fits should reflect lower variability, indicating that her refinement does not align with the observed data trends.