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------------------------------------------------ Beginning in the middle of summer, the average temperature for a certain year in Phoenix could approximately be modeled by the function

[tex]f(x)=26 \cos (0.0055 \pi x)+89[/tex],

where [tex]x[/tex] represents the number of days since the middle of the summer, and [tex]f(x)[/tex] represents the average temperature in degrees Fahrenheit.

What was the lowest average temperature in Phoenix that year (approximately)?

A. [tex]63^{\circ} F[/tex]
B. [tex]26^{\circ} F[/tex]
C. [tex]55^{\circ} F[/tex]
D. [tex]89^{\circ} F[/tex]

To find the lowest average temperature in Phoenix that year, we'll examine the function [tex]$ f(x) = 26 \cos(0.0055 \pi x) + 89 $[/tex], which models the temperature.

Here's a step-by-step guide:

1. Understand the role of cosine: The function [tex]$ \cos(0.0055 \pi x) $[/tex] is crucial here. The cosine function has values ranging between -1 and 1.

2. Identify how the amplitude affects the temperature: Since [tex]$ f(x) = 26 \cos(0.0055 \pi x) + 89 $[/tex], the term [tex]$ 26 \cos(0.0055 \pi x) $[/tex] means the temperature fluctuates 26 degrees above and below 89. Therefore, the amplitude of the cosine function, which is 26, indicates how much the temperature varies from the midline value of 89.

3. Determine the lowest value of cosine: The minimum value that [tex]$ \cos(0.0055 \pi x) $[/tex] can take is -1. This is a property of the cosine function.

4. Calculate the lowest average temperature:
[tex]\[
\text{Minimum of } f(x) = 26 \times (-1) + 89 = -26 + 89 = 63
\][/tex]

Therefore, the lowest average temperature in Phoenix that year was approximately [tex]$ 63^\circ F $[/tex].