Someone please help me!!! I think I've found the correct formula for this but I don't understand exactly why this is a sine function and how to find the days of the year when Portland has 11 and 15 hours of daylight. PLEASE HELP!!

Question

The equation I got was: D = 4 sin(2π t / 365) + 13. Is this right??

It is possible to use a sinusoidal function to model the amount of perceived daylight in a certain location over the course of a year. For Portland, Oregon, there is a minimum of 9 hours of “daylight” on the 1st day of winter and a maximum of 17 hours of “daylight” on the 1st day of summer. Let D represent the number of hours of “daylight” in Portland, Oregon, T days after the 1st day of spring (assume that T = 0 corresponds to March 20th). You may assume that 1 year has 365 days.
Find a formula for such a function, being sure to explain the practical meanings of any important pieces of the formula (amplitude,midline, and period). Use your formula to determine on what days of the year (month and day, not just T’s value) Portland has about 11 hours of “daylight” and about 15 hours of “daylight”. Please round to the nearest day, if not exact.

How do I solve the function I got for t?? I'm assuming I should plug in 11 and 15 for D but I don't know how to solve it from there and how to get an exact date.
Also, I'm not sure what the practical meanings of the period, amplitude, and midline are... Can someone explain them to me please?? Thank you!!

Answer 1

Ok. You have your function

D = 4 sin(2π t / 365) + 13

where D represent the number of hours of “daylight” in Portland

You want to find when D=11. So, solve

4 sin(2π t / 365) + 13 = 11
4 sin(2π t / 365) = -2
sin(2π t / 365) = -1/2

Now, sin(7π/6) = -1/2, so
2πt/365 = 7π/6
t = 7/6 * 365/2 = 212.9

So, you want the date 213 days after March 20.

Pulling out your handy Julian date calendar, you find that March 20 is day 79 of the year, so you want day 292, or Oct 19.

Do similar work for D=15. Or, considering the symmetric nature of sin(x), you can get the date another way.

Answer 2

That makes so much more sense now. Thank you!! So I did D = 15 and got 30.4. So 30 days after March 20 is April 19. Is that right?? Also, do you know what the practical meanings of the amplitude, period and midline are??? I'm assuming the midline is kind of the average hours of sunlight over the course of the year, but I'm not sure how to explain the amplitude and period. Is the midline the average hours of sunlight, and the amplitude the growth or decline of hours of sunlight? Really not sure about the period though. Thanks again!!

Answer 3

yes, the midline is the average hours of daylight.

The amplitude is the maximum deviation above or below that average.

Answer 4

So what about the period?? Would that be just the year throughout which the amount of sunlight changes???

Answer 5

the period is how long it takes for the values to repeat. Naturally, the period is one year, or 365 days.

Recall that the period of sin(x) is 2pi
for any value of x, sin(x+2pi) = sin(x)