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Trigonome tric Graphs
Word problem involving a sine or cosine function: Problem type 2
Jameson
Español
The average monthly temperature changes from month to month. Suppose that, for a given city, we can model the average temperature in a month with the following function.

f(t)=60-27 sin((pi)/(6)t)
In this equation, 
f(t) is the average temperature in a month in degrees Fahrenheit, and 
t is the month of the year (January 
=1, February=2, 
dots ).
Find the following. If necessary, round to the nearest hundredth.
Time for one full cycle of 
f 
◻ months
Number of cycles of 
f per month: 
◻
Minimum average temperature in a month: 
◻ 
◻ ० Fahrenheit
Explanation
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2:21 PM
4/22/2024

Import favorites\newlineMVNU Students Ho...\newlineA\newlineALEKS - Jameson C...\newlineTrigonome tric Graphs\newlineWord problem involving a sine or cosine function: Problem type 22\newlineJameson\newlineEspañol\newlineThe average monthly temperature changes from month to month. Suppose that, for a given city, we can model the average temperature in a month with the following function.\newlinef(t)=6027sin(π6t) f(t)=60-27 \sin \left(\frac{\pi}{6} t\right) \newlineIn this equation, f(t) f(t) is the average temperature in a month in degrees Fahrenheit, and t t is the month of the year (January =1 =1 , February=22, \ldots ).\newlineFind the following. If necessary, round to the nearest hundredth.\newlineTime for one full cycle of f f \square months\newlineNumber of cycles of f f per month: \square \newlineMinimum average temperature in a month: \square \square Fahrenheit\newlineExplanation\newlineCheck\newline(C) 20242024 McGraw Hill LLC. All Rights Reserved.\newlineTerms of Use\newlinePrivacy Center\newlineAccessibility\newlineType here to search\newlinet t 00\newline(11))\newline22:2121 PM\newline44/2222/20242024

Full solution

Q. Import favorites\newlineMVNU Students Ho...\newlineA\newlineALEKS - Jameson C...\newlineTrigonome tric Graphs\newlineWord problem involving a sine or cosine function: Problem type 22\newlineJameson\newlineEspañol\newlineThe average monthly temperature changes from month to month. Suppose that, for a given city, we can model the average temperature in a month with the following function.\newlinef(t)=6027sin(π6t) f(t)=60-27 \sin \left(\frac{\pi}{6} t\right) \newlineIn this equation, f(t) f(t) is the average temperature in a month in degrees Fahrenheit, and t t is the month of the year (January =1 =1 , February=22, \ldots ).\newlineFind the following. If necessary, round to the nearest hundredth.\newlineTime for one full cycle of f f \square months\newlineNumber of cycles of f f per month: \square \newlineMinimum average temperature in a month: \square \square Fahrenheit\newlineExplanation\newlineCheck\newline(C) 20242024 McGraw Hill LLC. All Rights Reserved.\newlineTerms of Use\newlinePrivacy Center\newlineAccessibility\newlineType here to search\newlinet t 00\newline(11))\newline22:2121 PM\newline44/2222/20242024
  1. Calculate period: To find the time for one full cycle, we need to determine the period of the sine function. The period of a sine function sin(Bt)\sin(Bt) is (2π)/B(2\pi)/B. Here, BB is (π/6)(\pi/6).
  2. Determine cycles per year: Calculate the period: (2π)/(π/6)=2π×(6/π)=12(2\pi)/(\pi/6) = 2\pi \times (6/\pi) = 12.
  3. Find minimum temperature: The time for one full cycle of ff is 1212 months.
  4. Find minimum temperature: The time for one full cycle of ff is 1212 months.To find the number of cycles of ff per year, we divide 1212 months by the period of the function. Since the period is 1212 months, there is 11 cycle per year.
  5. Find minimum temperature: The time for one full cycle of ff is 1212 months.To find the number of cycles of ff per year, we divide 1212 months by the period of the function. Since the period is 1212 months, there is 11 cycle per year.For the minimum average temperature, we look at the sine function. The minimum value of sin(π6t)\sin\left(\frac{\pi}{6}t\right) is 1-1. So, we plug 1-1 into the function for sin(π6t)\sin\left(\frac{\pi}{6}t\right).
  6. Find minimum temperature: The time for one full cycle of ff is 1212 months.To find the number of cycles of ff per year, we divide 1212 months by the period of the function. Since the period is 1212 months, there is 11 cycle per year.For the minimum average temperature, we look at the sine function. The minimum value of sin(π6t)\sin\left(\frac{\pi}{6}t\right) is 1-1. So, we plug 1-1 into the function for sin(π6t)\sin\left(\frac{\pi}{6}t\right).Calculate the minimum temperature: 121200.

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