On the Arctic Circle, there are 24 hours of daylight on the...

On the Arctic Circle, there are 24 hours of daylight on the summer solstice, June 21 (the 173rd day of the year), and 24 hours of darkness on the winter solstice, December 21 (day -10 and day 355 ). There are 12 hours of daylight and 12 hours of darkness midway between the summer and winter solstices (days 81 and 264 ). a. What are the maximum and minimum hours of daylight in the Arctic Circle? Use these values to calculate the amplitude and average value of the hours of daylight cycle. b. Calculate the period and horizontal shift of the hours of daylight cycle. Use these values to calculate the parameters $b$ and $c$ for a model of the form $f(x)=a \sin (b x+c)+d$ c. Use the constants from parts $a$ and $b$ to construct a sine model for the hours of daylight in the Arctic Circle.

On the Arctic Circle, there are 24 hours of daylight on the summer solstice, June 21 (the 173rd day of the year), and 24 hours of darkness on the winter solstice, December 21 (day -10 and day 355 ). There are 12 hours of daylight and 12 hours of darkness midway between the summer and winter solstices (days 81 and 264 ).
a. What are the maximum and minimum hours of daylight in the Arctic Circle? Use these values to calculate the amplitude and average value of the hours of daylight cycle.
b. Calculate the period and horizontal shift of the hours of daylight cycle. Use these values to calculate the parameters $b$ and $c$ for a model of the form $f(x)=a \sin (b x+c)+d$
c. Use the constants from parts $a$ and $b$ to construct a sine model for the hours of daylight in the Arctic Circle.

Key Concepts

Sine Function Modeling

Modeling with a sine function involves constructing an equation of the form f(x) = a sin(bx + c) + d, where 'a' is the amplitude, 'b' is related to the period (with the period being 2?/b), 'c' represents the horizontal phase shift, and 'd' is the vertical shift. This approach is useful for representing periodic phenomena such as variations in daylight hours over the course of a year.

Horizontal Shift (Phase Shift)

The horizontal shift, or phase shift, indicates the offset of the sine function along the horizontal axis. In a daylight hours model, it aligns the sine curve with known days (such as solstices) where maximum or minimum daylight occurs, ensuring the model accurately reflects the timing of seasonal changes.

Period

The period of a sine function is the length of one complete cycle of the function. For a seasonal daylight cycle, this represents the time it takes for the cycle of increasing and decreasing daylight hours to repeat, typically one year (365 days), which defines the cyclic nature of seasonal change.

Amplitude

Amplitude in a sine function represents half the distance between the maximum and minimum values of the function. In the context of daylight hours, it is calculated by taking half the difference between the longest and shortest day lengths, reflecting how much the daylight hours vary seasonally.

Vertical Shift (Average Value)

The vertical shift or average value is the midpoint between the maximum and minimum of the sine function. For daylight hours, it is calculated by averaging the longest and shortest day lengths, indicating the constant center around which the hours of daylight oscillate over a year.

Transcript

00:01 So in this video we're going to discuss basically everything involving periodic motion.

00:05 Also, you know, sinusoidal functions or cyclic functions and these different models that we have.

00:11 So let's just look at a stereotypical basic equation.

00:14 We'll call it f of x.

00:17 And really, we have a choice between sine and cosine.

00:20 We're going to go with sign, but really we'll see that they behave very similarly.

00:25 And we can actually modify sign functions to b cosine functions and the other way around as well.

00:30 So what we have is a times the sign of bx plus c plus d.

00:41 So what we end up seeing is this is the general form for our sinusoidal functions or our cyclic functions.

00:48 And this is going to be how we model all kinds of equations.

00:52 So the first one i want to point out, we'll just let everything kind of be normal.

00:57 But the main thing we want to point out is right here, the amplitude.

01:01 Notice that as this gets bigger, the height of the function gets bigger.

01:06 That's going to be the amplitude.

01:08 So we see that with a traditional function, where this equals zero and this equals zero, that our amplitude goes to plus one and minus one.

01:17 So this means we have an amplitude of one.

01:20 However, if this got brought up to two, we would now have plus two minus two.

01:25 That's an amplitude of two, or an amplitude of two, and the average value of the function would be zero.

01:32 Now let's take a look at what the b value does in terms of actually we'll look at the d value.

01:39 The b value is very simple...

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