Monthly Temperatures in Charleston The monthly normal mean...

Monthly Temperatures in Charleston The monthly normal mean temperatures for the last 30 years in Charleston, SC, are shown in Table $4.3 .$ A scatter plot suggests that the mean monthly temperatures follow a sinusoidal curve over time. Assume that the sinusoid has equation $y=a \sin (b(t-h))+k$ (a) Given that the period is 12 months, find $b$ . (b) Assuming that the high and low temperatures in the table determine the range of the sinusoid, find $a$ and $k$ (e) Find a value of $h$ that will put the minimum at $t=1$ and the maximum at $t=7$ . (d) Superimpose a graph of your sinusoid on a scatter plot of the data. How good is the fit? (e) Use your sinusoidal model to predict dates in the year when the mean temperature in Charleston will be $70^{\circ} .$ (Assume that $t=0$ represents January $1 . )$

Monthly Temperatures in Charleston The monthly normal mean temperatures for the last 30 years in
Charleston, SC, are shown in Table $4.3 .$ A scatter plot suggests that the mean monthly temperatures follow a sinusoidal curve over time. Assume that the sinusoid has equation $y=a \sin (b(t-h))+k$
(a) Given that the period is 12 months, find $b$ .
(b) Assuming that the high and low temperatures in the table determine the range of the sinusoid, find $a$ and $k$
(e) Find a value of $h$ that will put the minimum at $t=1$ and the maximum at $t=7$ .
(d) Superimpose a graph of your sinusoid on a scatter plot of the data. How good is the fit?
(e) Use your sinusoidal model to predict dates in the year
when the mean temperature in Charleston will be $70^{\circ} .$ (Assume that $t=0$ represents January $1 . )$

Key Concepts

Sinusoidal Functions

Sinusoidal functions model periodic phenomena using sine or cosine curves. They are characterized by smooth, repetitive oscillations and are commonly used to represent cyclical patterns in various fields such as physics, engineering, and environmental science.

Period and Frequency

The period of a sinusoidal function is the interval over which the function completes one full cycle. It is inversely related to the frequency parameter (often represented by b in the equation y = a sin(b(t - h)) + k), where the period T is given by T = 2? / b. This concept is essential for matching a model to a phenomenon that repeats at regular intervals.

Amplitude

Amplitude is the measure of the maximum deviation of the sinusoidal function from its midline (the vertical shift). It represents the height of the peaks or the depth of the troughs and is calculated as half the difference between the maximum and minimum values of the function. This parameter is crucial for understanding the magnitude of the oscillations.

Vertical Shift

The vertical shift is the constant that moves the sinusoidal function up or down on the graph, setting the midline around which the oscillations occur. It represents the average value or baseline of the data being modeled and is critical in aligning the function with real-world measurements.

Phase Shift

Phase shift refers to the horizontal displacement of the sinusoidal function along the time axis. This parameter (often denoted by h) adjusts where the sine or cosine wave starts, ensuring that key features of the model, such as peaks and troughs, align with the corresponding features in the data.

Model Fitting and Prediction

Model fitting involves adjusting the parameters of a sinusoidal function to align it closely with observed data, thereby creating a mathematical model of the periodic behavior. Once a well-fitted model is established, it can be used to predict future values by computing the function at given time points, highlighting the practical applications of these mathematical tools.

Transcript

00:04 All right, we're dealing with the temperatures that they gave us in table 4 .3.

00:09 And if you do a scatter plot of this, we end up with there, the low being january or month number one.

00:18 I'm assuming that would be january.

00:20 And then we have month number 12, december there.

00:23 And then if we just follow this pattern, we're going to continue with a sinusoidal type of graph here.

00:31 So this 1 to 12, that's going to give us our period for the graph.

00:41 And so that means we are looking at, in part a, finding the value of b in our equation.

00:48 Well, the period equals 2 pi divided by b in our general equation.

00:55 So multiply both sides by b.

00:59 We get 12b equals 2 pi, divide by the 12, and we end up with b equaling pi over 6.

01:07 Now, the high and low temperatures in the table are going to give us the range of our function here.

01:18 So our low temperature is at 48 degrees, and our high temperature being at 82 degrees.

01:28 So for part b, we are going to find our amplitude by doing the maximum value minus the minimum value divided by 2.

01:41 82 minus 488 divided by 2 gives us an amplitude of 17.

01:48 And then we have a midline here in our graph, and that location of the midline is the value of k.

02:00 That's the amount of our vertical shift.

02:03 And so that's simply going to be our amplitude below the maximum or above the minimum.

02:09 So 82 minus 17 or 48 plus the 17.

02:16 I'm going to do the 82 minus 17 gives us a value of k of 65 degrees.

02:30 And then for part c, find a value of h that will put the minimum at t equals 1 and the maximum at t equals 7.

02:38 Okay, well, we want our minimum here at 1.

02:40 And normally when you're dealing with sign, which is what they want us trying to use, normally we start at the origin and go up from there.

02:54 So that means that the origin would be in the middle of our maximum and minimum.

03:01 So our value of h is going to be the midpoint of 7 and 1.

03:08 So 7 plus 1 over 2 gives us an h -bile.

03:11 Of 4.

03:13 So we now have the equation of y equals or t for temperature equals whatever your variable is going to be, 17 times the sign of pi over 6 times the quantity t minus 4 close my parentheses in my brackets plus 65 and in part d you were supposed to take this equation and graph that on top of your scatter pot and you should see that it is a very good fit hitting most of the points.

03:55 Now for part e i'm supposed to use this model to predict when the temperature is going to be 70 degrees and we're assuming that our time t equals zero is january 1st...

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