How Does Daylight Vary by Month Using a Sinusoidal Function?

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The daylight hours at a specific location can be modeled by the equation h = 12 + 4sin[2π/12(m-9)], where m indicates the month. The shortest day of the year occurs when the sine function reaches its minimum value, which is -1, resulting in 8 hours of daylight. This model suggests the person lives in the Northern Hemisphere, as the minimum daylight occurs around December. The number 12 in the formula represents both the average daylight hours (12) and the annual cycle of months (12 months in a year). To find the minimum of a function, one typically looks for critical points where the derivative is zero, and for the sine function, its minimum value is -1.
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Daylight sinusoid any help appreciated :)

A person finds that the number of hours of daylight at his home is approximately given by

h = 12 + 4sin[2π/12(m-9)]
where m represents the number of the month (January = 1, February = 2, etc.)

a) At the person's location, how long is the shortest day of the year?

b) Does the person live in the Northern or Southern Hemisphere?

c) The number 12 occurs twice in the formula, but for two different reasons. Explain the real world meaning of each of the 12's.
 
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How does one usually find the minimum of a function?
 
Also, how does one find the minimum and maximum of sine in particular?
 

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I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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