Sine waves (precalc/calc/genius algebra students)?

Question

Hi all, I need help understanding how to graph sin waves.
An example problem is:

A lemonade stand sells a lot of drinks in the sumer and not very many in the winter. The best day of the year was August 10th (215th day), when 150 glasses were sold. The worst day was February 10th (41th day), when 40glasses were sold.

a) give the Domain, Range, and period of the situation above

b) Write an equation for a sine function that has been "danced" to fit the situation above [simplify]

c) use the equation from part b to find the # of glasses sold on June 1 (154th day)

d) write an equation for a cosine function that has been "danced' [simplified] to fit the situation above. Explain the similarities and differences between this cosine function and the sine function from part b




For (A), I got Domain to be all reals, and Range to be [40,150]. How do I solve for period?


How do I go about solving the rest of this from there? Thanks in advance for helping me understand this concept!

thanks a bunch cat, Sine graphs and sinusoids have been giving me trouble since the beginning of precalc

Okay, and to take that equation to cosine form, i just add (pi) to the x, so

55*sin(2pi/365(x+129+pi))+95

?

Answer 1

the domain is NOT all reals. It is from Julian Day (JD) 1 to JD 365+phase shift (see below). You only have so many days in a year.

The CYCLE is 365.days. This is being lazy, since it overlooks leap years, but who's perfect?

The first aspect of a wave is AMPLITUDE. For the sine, it goes from +1 to -1. Here, it goes from 40 to 150. So the mean is 95 and the extremes are plus and minus 55. So the wave is going to look like
Glasses(JD)= 95 + 55 Sin (2*pi/f [JD+phase shift]).
The 2*pi comes in since a simple sine function has a cycle of 2*pi radians. The frequency is the reciprocal of the time for the sine wave to go through 1 complete cycle, or 1/365 days.
The phase shift factor adjusts for the offset in between the real data and the sine factor for a given day, particularly day 1. The sine factor without a phase shift would have the mean value of 95 on day 1. According to the data, this actually happens on day (41+215)/2 or day 129. So we add this factor to indicate that we are 129 day further along in the sine factor than we should be.

So the final form is
Glasses(JD)= 95+55*Sin ((2*pi/365)[JD+129])

I call that a whole lot of DIRTY DANCING!!

Answer 2

2 ms is 2/1000 s = 0.002 s now is 0.002 smaller than 0.0025? ( of course) so there is less time for the wave to change in and it must be changing faster.