The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July.
a) Construct a function that models this behavior.
b) During what period is there more than 10 inches of snowfall?
I got the answers:
a) $6 + 5 \cos(\frac{\pi}{6} (1−x))$.
b) From November 23 to February 6.
But i don't how they calculate exactly to dates 23/11 -> 6/2. May someone explain to me? Thanks!
For question a), note that there are several (infinite) functions that behave as you described (see Ross Millikan comment). So, in order to solve this problem, we need further assumptions.
If we assume a periodic behaviour, the trigonometric functions are quite useful. If we further assume that the cosine (or the sine) function have the desired shape, then we should use it. I'll repeat myself by saying that there are other possible solutions. Perhaps the context of the original question will guide you to the cosine of the sine function.
So, assuming that the cosine is the function to be used, we can use the rest of the information of the question to build the answer. If the minimum snowfall is in July (month $7$), then the cosine should be minimum at that moment. Hence, the argument of the cosine should be $\pi$ when the month $x=7$. Furthermore, since the period is $12$ months, when we increase $x$ by $12$ months we should increase the argument of the cosine by $2 \pi$, which is equivalent of increasing by $\pi$ every $6$ months.
Since the average of the cosine over one period ($12$ months) is zero, and the average snowfall is $6$ inches, that gives us the independent coefficient. And since the minimum is $1$ inch, then the amplitude must be $5$ inches because $6 - 5 = 1$. Furthermore, we know that the maximum snowfall (in January) will be $6 + 5 = 11$ inches.
As for question b), we must solve $6 + 5 \cos (\frac{\pi}{6}(1-x)) = 10$, which gives in $[0; 12]$: $x = 2.229$ or $x = 11.771$. So, a bit after month $2$, February, and almost in the end of month $11$, November. To have the exact day, we must multiply the decimal part by the number of days in each month: $0.229 \times 28 = 6.4$ days and $0.771 \times 30 = 23.13$ days.