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The average monthly temperature, in degrees Fahrenheit, for a town in central Indiana can be modeled by the sinusoidal function T(m)=25.7 sin((pi)/(6)(m-4))+61.2, for 1 <= m <= 12 months. Based on the model, which of the following is true? (A) The maximum average monthly temperature is 61.2 degrees Fahrenheit. (B) The maximum average monthly temperature occurs at m=1 month. (C) The minimum average monthly temperature is 35.5 degrees Fahrenheit. (D) The minimum average monthly temperature occurs at m=7 months.

The average monthly temperature, in degrees Fahrenheit, for a town in central Indiana can be modeled by the sinusoidal function T(m)=25.7sin⁡(π6(m−4))+61.2 T(m)=25.7 \sin \left(\frac{\pi}{6}(m-4)\right)+61.2 , for 1≤m≤12 1 \leq m \leq 12 months. Based on the model, which of the following is true?\newline(A) The maximum average monthly temperature is 6161.22 degrees Fahrenheit.\newline(B) The maximum average monthly temperature occurs at m=1 m=1 month.\newline(C) The minimum average monthly temperature is 3535.55 degrees Fahrenheit.\newline(D) The minimum average monthly temperature occurs at m=7 m=7 months.

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Q. The average monthly temperature, in degrees Fahrenheit, for a town in central Indiana can be modeled by the sinusoidal function T(m)=25.7sin⁡(π6(m−4))+61.2 T(m)=25.7 \sin \left(\frac{\pi}{6}(m-4)\right)+61.2 , for 1≤m≤12 1 \leq m \leq 12 months. Based on the model, which of the following is true?\newline(A) The maximum average monthly temperature is 6161.22 degrees Fahrenheit.\newline(B) The maximum average monthly temperature occurs at m=1 m=1 month.\newline(C) The minimum average monthly temperature is 3535.55 degrees Fahrenheit.\newline(D) The minimum average monthly temperature occurs at m=7 m=7 months.
  1. Analyze Function: Analyze the given sinusoidal function for temperature.\newlineThe function T(m)=25.7sin⁡(π6(m−4))+61.2T(m) = 25.7 \sin\left(\frac{\pi}{6}(m-4)\right) + 61.2 models the average monthly temperature in degrees Fahrenheit. The general form of a sinusoidal function is T(m)=Asin⁡(B(m−C))+DT(m) = A \sin(B(m - C)) + D, where:\newline- AA is the amplitude (half the distance between the maximum and minimum values),\newline- BB affects the period of the function,\newline- CC is the horizontal shift (which determines where the maximum or minimum occurs),\newline- DD is the vertical shift (which determines the midline of the function).
  2. Determine Amplitude: Determine the amplitude of the function.\newlineThe amplitude is given by the coefficient of the sine function, which is 25.725.7. This means the temperature fluctuates 25.725.7 degrees above and below the midline.
  3. Determine Midline: Determine the midline of the function.\newlineThe midline is given by the constant term added to the sine function, which is 61.261.2. This is the average value around which the temperature fluctuates.
  4. Calculate Maximum Temperature: Calculate the maximum average monthly temperature. The maximum temperature occurs at the midline plus the amplitude. Therefore, the maximum average monthly temperature is 61.2+25.7=86.961.2 + 25.7 = 86.9 degrees Fahrenheit.
  5. Calculate Minimum Temperature: Calculate the minimum average monthly temperature. The minimum temperature occurs at the midline minus the amplitude. Therefore, the minimum average monthly temperature is 61.2−25.7=35.561.2 - 25.7 = 35.5 degrees Fahrenheit.
  6. Determine Maximum Month: Determine when the maximum average monthly temperature occurs.\newlineThe maximum temperature occurs when the sine function is at its peak, which is at sin⁡(π2)\sin(\frac{\pi}{2}). To find the corresponding month (mm), we set the inside of the sine function equal to π2\frac{\pi}{2} and solve for mm:\newline(π6)(m−4)=π2\left(\frac{\pi}{6}\right)(m-4) = \frac{\pi}{2}\newlinem−4=3m - 4 = 3\newlinem=7m = 7\newlineTherefore, the maximum average monthly temperature occurs at m=7m=7 months.
  7. Determine Minimum Month: Determine when the minimum average monthly temperature occurs.\newlineThe minimum temperature occurs when the sine function is at its lowest, which is at sin⁡(3π2)\sin(\frac{3\pi}{2}). To find the corresponding month (mm), we set the inside of the sine function equal to 3π2\frac{3\pi}{2} and solve for mm:\newlineπ6(m−4)=3π2\frac{\pi}{6}(m-4) = \frac{3\pi}{2}\newlinem−4=9m - 4 = 9\newlinem=13m = 13\newlineHowever, since the model is only valid for 1≤m≤121 \leq m \leq 12, we need to find the equivalent value within this range. Since the sine function has a period of 1212 months, we subtract 1212 from 1313 to find the equivalent month within the valid range:\newlinem=13−12m = 13 - 12\newlinem=1m = 1\newlineTherefore, the minimum average monthly temperature occurs at mm00 month.

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