MATH QUESTION PLSSSS HELP

Question

The peak of the full moon occurs, on average, every 29.53 days, and the approximate value of the moon that is lit can be modeled by a sinusoid. Assume that all of the moon is lit at the peak of the full moon and that none of the moon is lit at the apex of the new moon.

If the peak of the full moon was at 8 PM on April 30, what decimal value, 

D

, of the moon will be lit at midnight on June 10 of the same year?

D=?

Answer choices

A. 0.1098

B. 0.1075

C. 0.1038

D. 0.1067

E. None of the above

Answer 1

The equation for a sinusoid is y = a*sin( k(x-d) ) + h

The period of the sine wave in this example is 29.53 days since it takes 29.53 days to go from one peak to the next. k is always 2π/period so k = 2π/29.53.

Next we have to get a and h. We know that the highest possible value of the sine graph is 1 (when the moon is full) and the lowest possible value is 0 (when it's a new moon). Therefore we would want to raise the sine graph up 1/2 so the apex and give the function an altitude of 1/2. Therefore the peak would rise to 1 and the crest would drop to 0. Generally speaking h + a = peak and h - a = crest. So both a and h are 0.5.

So far we have y = 0.5sin ( (2π/29.53) (x - d) ) + 0.5

To get d plug in a point. We know on April 30 (We'll call that day 0), we had a full moon (which means y = 1)

Therefore we have the point (0,1). Plug this in and solve for d.

January 10 on midnight is 41 days and 4 hours later or 41.166667.

Therefore on this day at this time, x = 41.166667.

Plugging this in gives y = 0.1067

We get d = -29.53/4

So the equation is y = 0.5sin ( (2π/29.53) (x + 29.53/4) ) + 0.5