Math Question (Sinusoidal function)

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 November 8, what decimal value, , of the moon will be lit at midnight on December 14 of the same year?

A.

B.

C.

D.

E. None of the above.

Answer 1

If you consider 100% illumination at 8pm on Nov 8, then a cosine function does well at modeling this since cos(0) = 1

The generic cosine function is y = Acos[B(x - C)] + D where A is the amplitude, B is used to calculate the period where period = 2π/B, C is the phase shift or horizontal shift, and D is the vertical shift.

In this case, since amplitude goes from 0% to 100%, we can consider A = 50% (or 0.5) and D = 0.5. That way the function will oscillate between 100% (1.0) and 0% (0.0).

There is no horizontal shift since we are using a cosine function and we want the max value at t = 0. We can allow t to be in days and we'll need to use fractional days to accommodate the hours.

The value of B is determined from 29.53 = 2π/B therefore B = 0.212773

So our function is y = 0.5cos(0.212773t) + 0.5

Between Nov 8 and Dec 14, there are 36 days (22 days in November between Nov 8 and Nov 30 and 14 days in Dec) and we need to add another 4 hours for the time of day differential so 4/24 or 1/6 of a day. So t = 36.166667

So since y(t) = 0.5cos(0.212773t) + 0.5 then y(36.166667) = 0.5cos(0.212773•(36.166667)) + 0.5

I'll let you do the rest. If using a calculator, make sure you are set in radians.