Math  /  Data & Statistics

Question15. Average monthly temperatures for the city of Hamilton, Ontario, from Januáry to December are shown. \begin{tabular}{|c|c|} \hline Month & Temperature (C)\left({ }^{\circ} \mathrm{C}\right) \\ \hline 1 & -4.8 \\ \hline 2 & -4.8 \\ \hline 3 & -0.2 \\ \hline 4 & 6.6 \\ \hline 5 & 12.7 \\ \hline 6 & 18.6 \\ \hline 7 & 21.9 \\ \hline 8 & 20.7 \\ \hline 9 & 16.4 \\ \hline 10 & 10.5 \\ \hline 11 & 3.6 \\ \hline 12 & -2.3 \\ \hline \end{tabular} a) Make a scatter plot of the data. b) Use the table and the graph to write a sinusoidal function to model the data. c) Graph your model on the same set of axes as your scatter plot. Comment on the accuracy of the fit.

Studdy Solution

STEP 1

What is this asking? We need to graph the temperatures, then find a *wave-like* equation that *best fits* those temperatures and graph that equation too! Watch out! Remember that sinusoidal functions can be tricky.
Make sure to double-check your calculations for the amplitude, period, phase shift, and vertical shift.

STEP 2

1. Plot the temperatures.
2. Find the sinusoidal function.
3. Graph the function.

STEP 3

Let's **plot** these temperatures!
We'll put the month number on the horizontal axis (that's our *x*) and the temperature on the vertical axis (that's our *y*).
So, for example, January is month 1, and its temperature is 4.8-4.8^\circ C, giving us the point (1,4.8)(1, -4.8).

STEP 4

The **amplitude** is *half the difference* between the highest and lowest temperature.
The highest temperature is 21.921.9^\circ C in July (month 7) and the lowest is 4.8-4.8^\circ C in January and February (months 1 and 2).
So, our calculation is: 21.9(4.8)2=21.9+4.82=26.72=13.35 \frac{21.9 - (-4.8)}{2} = \frac{21.9 + 4.8}{2} = \frac{26.7}{2} = 13.35 So, our **amplitude** is 13.35\bf{13.35}.

STEP 5

The **vertical shift** is the *midpoint* of the highest and lowest temperatures.
We can find this by *averaging* the highest and lowest temperatures: 21.9+(4.8)2=17.12=8.55 \frac{21.9 + (-4.8)}{2} = \frac{17.1}{2} = 8.55 So, our **vertical shift** is 8.55\bf{8.55}.

STEP 6

Since temperature repeats every 12 months, our **period** is 12\bf{12}.
The formula that relates the period to the *b*-value in our sinusoidal function is Period=2πb \text{Period} = \frac{2\pi}{b} .
Let's solve for *b*: 12=2πb 12 = \frac{2\pi}{b} b=2π12=π6 b = \frac{2\pi}{12} = \frac{\pi}{6} So, our *b*-value is π6\bf{\frac{\pi}{6}}.

STEP 7

We'll use a cosine function.
Cosine starts at its maximum value.
Our maximum temperature is in July, which is month 7.
This means our graph is shifted 7 units to the right.
The **phase shift** is 7\bf{7}.
We can represent this shift in our equation as (x7)(x - 7).

STEP 8

Putting it all together, our sinusoidal function is: y=13.35cos(π6(x7))+8.55 y = 13.35\cos\left(\frac{\pi}{6}(x - 7)\right) + 8.55

STEP 9

Now, graph this function on the same graph as your scatter plot.
You should see that the curve *fits the data pretty well*!
There might be some small differences, but that's okay.
Real-world data is rarely perfectly sinusoidal.

STEP 10

The sinusoidal function that models the data is y=13.35cos(π6(x7))+8.55 y = 13.35\cos\left(\frac{\pi}{6}(x - 7)\right) + 8.55 and when graphed, it's a pretty good fit to the temperature data!

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