IB MATH AI SL, Paper 2, May, 2022, TZ1, Solved Past Paper

Boris has been observing the number of daylight hours on the first day of each month in a town located in the northern hemisphere. He has plotted these observations on a scatter plot, which is depicted in the diagram below.

The curve in the diagram is represented by the function $y=f(t)$, where $t$ denotes the time in months since Boris began recording these values. Boris hypothesizes that $f(t)$ could be modeled using a quadratic function.

However, there are considerations to be made regarding the suitability of a quadratic function for modeling daylight hours over time. Quadratic functions are characterized by their parabolic shape, which implies that they have a single maximum or minimum point. This characteristic may not align well with the cyclical nature of daylight hours, which typically follow an annual cycle with both a maximum and a minimum each year.

Paula proposes an alternative model for the daylight hours, suggesting a trigonometric function of the form $f(t)=a\cos (bt)+d$, where $t\ge 0$. This model is more suitable for periodic phenomena like daylight hours, as it can accommodate the regular oscillations observed throughout the year.