Well, if I had to model monthly temperatures in Atlanta, I'd do it with a smile! Let's dig in and have some fun with math, shall we?
To find the amplitude, we need to identify the difference between the highest and lowest points of the sinusoidal function. In this case, it goes from 32 to 77, so the amplitude is 77 - 32 = 45.
Now, let's calculate the period. The period is the length it takes for the sinusoidal function to complete one full cycle. Since we're working with months as our x-axis unit, the period would be 12 months.
Next up, we'll determine the phase shift. The phase shift is the horizontal shift of the sinusoidal function compared to the standard sine or cosine function. In this case, since x = 1 represents January, there is no horizontal shift, so the phase shift is 0.
Finally, we'll look at the vertical shift, which refers to the vertical displacement of the function. In this case, it seems that the average temperature is centered around 55 degrees. Therefore, the vertical shift is 55.
So, the equation of the sinusoidal function that models the monthly temperatures in Atlanta would be:
T(x) = 45sin((2Ο/12)x) + 55
Now, it's time to put on our clown noses and plug in the numbers for July and December!
For July, x would be 7 since it represents the seventh month. Plugging it into our equation, we get:
T(7) = 45sin((2Ο/12) * 7) + 55
T(7) = 45sin((7Ο/6)) + 55
Calculating that, we find that Atlanta's average temperature in July according to our model would be approximately 73 degrees.
For December, x would be 12, the twelfth month. Plugging it into our equation, we get:
T(12) = 45sin((2Ο/12) * 12) + 55
T(12) = 45sin((2Ο)) + 55
Since the sine function is periodic with a period of 2Ο, we know that sin(2Ο) = 0. Therefore:
T(12) = 45 * 0 + 55
T(12) = 55
According to our model, Atlanta's average temperature in December would be approximately 55 degrees.
And there you have it! A bit of math and a bit of humor to model Atlanta's temperatures and predict what July and December might bring. Stay cool and warm, my friend!