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!!

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.

Also, I'm not sure how to explain the practical meanings of the pieces of the formula like the amplitude, midline, and period. Can someone explain that to me too?? THANK YOU!!

your equation is right. Maybe you are not as confused as you think.

Real helpful Steve. *eyeroll*

Yes, the equation you have derived is correct. The formula you obtained, D = 4 sin(2π t / 365) + 13, can be used to model the amount of perceived daylight in Portland, Oregon throughout the year.

Let's break down the formula:

D represents the number of hours of "daylight" in Portland, Oregon.
t represents the number of days after the 1st day of spring (March 20th).

The formula is a sinusoidal function, with a sine function as the main component. The sine function is responsible for the periodic variations in daylight hours throughout the year.

Here are the important components of the formula:

Amplitude: The amplitude of a sinusoidal function is the maximum distance from the midline to the peak or trough. In this case, the amplitude is 4, which means the difference between the maximum and minimum amount of daylight in Portland is 4 hours.

Midline: The midline represents the horizontal shift of the sine function. In this case, the midline is 13, which means the average amount of daylight in Portland throughout the year is 13 hours.

Period: The period of a sinusoidal function is the length of one complete cycle. In this case, the period is 365, representing the number of days in a year.

To find the days when Portland has about 11 hours of daylight, you can set D = 11 in the equation and solve for t:

11 = 4 sin(2π t / 365) + 13

Rearranging the equation, we get:

4 sin(2π t / 365) = -2

Next, divide both sides by 4:

sin(2π t / 365) = -1/2

To find the corresponding days, you will need to find the angles whose sine is -1/2. These angles can be found using the inverse sine function or by referring to the unit circle.

Sine is negative in two quadrants: the third (between 180 and 270 degrees) and the fourth (between 270 and 360 degrees). In both quadrants, the sine function has the same value, -1/2.

For the third quadrant, the angle is approximately 193.11 degrees or around 3.36 radians. For the fourth quadrant, the angle is approximately 346.89 degrees or around 6.04 radians.

Using the formula:

t = (angle * 365) / (2π)

Calculating for the third quadrant angle, t ≈ (3.36 * 365) / (2π) ≈ 189.33 days.

Calculating for the fourth quadrant angle, t ≈ (6.04 * 365) / (2π) ≈ 337.67 days.

Therefore, Portland has about 11 hours of daylight on approximately the 189th and 338th day of the year.

Similarly, you can solve for the days when Portland has about 15 hours of daylight by setting D = 15 in the equation and following the same process.

Please note that the results may not be exact due to rounding and the assumption of a 365-day year.