Temperature changes are observed over the course of three summer months and modeled by the function f (x) = 3 cosine (StartFraction pi Over 5 EndFraction x + StartFraction pi Over 5 EndFraction) + 25, where x represents the days after June 1 and f(x) represents the temperature in degrees Celsius on day x. The first occurrence of the highest temperature in the cycle is day with a temperature of °C. The first occurrence of the lowest temperature in the cycle is day with a temperature of °C.

The first occurrence of the highest temperature and lowest temperature in the cycle are days with temperatures of 28°C and 22°C, respectively

The first highest and least temperatures

The function is given as:

[tex]f\left(x\right)=3\cos\left(\frac{\pi}{5}x\ +\ \frac{\pi}{5}\right)\ +\ 25[/tex]

As a general rule, we have:

[tex]\cos(\theta) = -1[/tex] --- minimum

[tex]\cos(\theta) = 1[/tex] --- maximum

This means that the highest value and the lowest value of [tex]\cos\left(\frac{\pi}{5}x\ +\ \frac{\pi}{5}\right)[/tex] are 1 and -1, respectively.

So, we have:

Highest temperature

[tex]f\left(x\right)=3\cos\left(\frac{\pi}{5}x\ +\ \frac{\pi}{5}\right)\ +\ 25[/tex]

[tex]f\left(x\right)=3 * 1\ +\ 25[/tex]

[tex]f\left(x\right)=28[/tex]

Lowest temperature

[tex]f\left(x\right)=3\cos\left(\frac{\pi}{5}x\ +\ \frac{\pi}{5}\right)\ +\ 25[/tex]

[tex]f\left(x\right)=3 * -1\ +\ 25[/tex]

[tex]f\left(x\right)=22[/tex]

Hence, the first occurrence of the highest temperature and lowest temperature in the cycle are days with temperatures of 28°C and 22°C, respectively

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