Another Trigonometric Model Among Temperatures

Geometry Level 3

In 2015, students have followed some daily average temperature registrations in Vicenza, Veneto, Italy. The analysis of another data gathered in previous years convinced the scholars that the daily average temperature follows a trend schematically described by the following function family:

\[ \large f(t)=\displaystyle{\sin^2\left ( \dfrac{t-6}{4} \right )-\sin \left ( \dfrac{t-6}{4} \right ) \cos \left ( \dfrac{t-6}{4} \right )+a \cdot \cos^2 \left ( \dfrac{t-6}{4} \right )+b} \]

where \(t\) is time expressed in months of 30 days. For example, on 13 March 2015, 72th day of the year, time \(t\) is \(\dfrac{72}{30}=2.4\).

Find the value of parameters \(a\) and \(b\) knowing that on 13 March 2015 the average temperature was \( 11.01^\circ C\) and on 26 June of the same year was \( 32^\circ C\).

Using the parameters found previously, determine the days when the average temperature was higher than \(20^\circ C\) and lower than \(30^\circ C\).

(The temperature is measured in Celsius.)


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