Sales Sales of snowblowers are seasonal. Suppose the sales...

Sales Sales of snowblowers are seasonal. Suppose the sales of snowblowers in one region of the country are approximated by $$ S(t)=500+500 \cos \left(\frac{\pi}{6} t\right) $$ where $t$ is time in months, with $t=0$ corresponding to November. Find the sales for $(a)-(e)$ (a) November $\quad$ (b) January $\quad$ (c) February (d) May (e) August $\quad$ (f) Graph $y=S(t)$

Sales Sales of snowblowers are seasonal. Suppose the sales of snowblowers in one region of the country are approximated by
$$ S(t)=500+500 \cos \left(\frac{\pi}{6} t\right) $$
where $t$ is time in months, with $t=0$ corresponding to November. Find the sales for $(a)-(e)$
(a) November $\quad$ (b) January $\quad$ (c) February (d) May (e) August $\quad$ (f) Graph $y=S(t)$

Key Concepts

Periodic Functions

Periodic functions repeat their values at regular intervals or periods. In many real-world scenarios, phenomena such as seasonal behaviors or cyclic events are modeled using periodic functions, which help in understanding and predicting recurring patterns over time.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in modeling periodic behavior. They are used extensively in applications where oscillatory or cyclic patterns are present, allowing us to represent phenomena like sound waves, light waves, or seasonal variations in data.

Cosine Function Transformations

The cosine function can be transformed by adjusting its amplitude, period, and vertical shift, which tailors it to model specific scenarios accurately. In a transformed cosine function, the amplitude affects the height of the peaks and troughs, the period changes the distance between consecutive cycles, and the vertical shift moves the entire graph up or down.

Function Evaluation

Evaluating functions at specific points involves substituting given values into the function in order to determine the output. This process is essential for understanding how a function behaves at discrete time intervals or under particular conditions, enabling practical analysis and predictions.

Graphical Representation of Functions

Graphing functions provides a visual representation of how the function behaves over a range of inputs. It helps in identifying key features such as maximum and minimum values, periodicity, and trends, thereby offering an intuitive understanding of the underlying mathematical relationship.

Seasonal Modeling

Seasonal modeling uses mathematical functions to capture and predict variations due to seasons or recurring time-based patterns. By employing periodic functions, such as the cosine function, one can model phenomena like sales cycles, temperature changes, or other seasonal effects, thereby aiding in planning and decision making.

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