Question

In a certain town, the longest day of the year, which is in june, lasts fourteen hours. the shortest day of the year, which is in december, lasts ten hours. twice per year, in march and september, the day is the same length as the night, or twelve hours. length of day varies sinusoidally through the year. write an equation for h(m, the length of the day in hours, as a function of the cosine of m, the number of months since january.

Answer 1

From the information given we can create a few data points:
(6,14), (3,12), (9,12) and (12,10). Where the x values are the months and the y values are the number of hours where is "day"

The answer will be in the form:

`y=A*cos(Bx+C)+D`

To get A, we have to find the range and then divided by 2. Therefore we have

`14-10=4*(1/2)=2`

**Since we are using cos we need to start and end at the same point and such, A is negative. Therefore


`A=-2`

D is 12 since is the middle value between 10 and 14.

C is zero.

To obtain the value of B we have:


`(2 \pi )/(B)=12`

When solved we have that


`B= ( \pi )/(6)`

Therefore the final equation is:


`y=-2cos(m* ( \pi )/(6))+12`