The average temperature, in °F, for Atlanta, GA is modeled by the sinusoidal function y = 18.25 sin(pi/6 t - 2.09) + 61.15 where t is the time in months and January is t=1.
What is the average Temperature for the month of February?
Using the same equation, what is the average temperature in Atlanta, GA, for the month of June?
plug in t=2 to get
18.25 sin(pi/6 * 2 - 2.09) + 61.15 = 45.38
for June, plug in t=6.
To find the average temperature for the month of February, substitute t=2 into the equation:
y = 18.25 sin(pi/6 * 2 - 2.09) + 61.15
Calculate the value inside the sine function:
(pi/6 * 2 - 2.09)
= (pi/3 - 2.09)
Now substitute this value back into the equation:
y = 18.25 sin(pi/3 - 2.09) + 61.15
Using a calculator, find the sine of (pi/3 - 2.09) and multiply by 18.25. Finally, add this result to 61.15 to get the average temperature for February.
To find the average temperature for the month of June, substitute t=6 into the equation:
y = 18.25 sin(pi/6 * 6 - 2.09) + 61.15
Calculate the value inside the sine function:
(pi/6 * 6 - 2.09)
= (pi - 2.09)
Now substitute this value back into the equation:
y = 18.25 sin(pi - 2.09) + 61.15
Using a calculator, find the sine of (pi - 2.09) and multiply by 18.25. Finally, add this result to 61.15 to get the average temperature for June.
To find the average temperature for a specific month, we need to substitute the value of "t" that corresponds to that month into the given equation.
For the month of February, we need to find the value of "t" for February. January is represented by "t = 1", so we can determine the value of "t" for each month by counting the number of months since January. Therefore, since February comes after January, the value of "t" for February will be 2.
Now, we substitute "t = 2" into the equation y = 18.25 sin(pi/6 t - 2.09) + 61.15:
y = 18.25 sin(pi/6 * 2 - 2.09) + 61.15
Simplifying further:
y = 18.25 sin(pi/3 - 2.09) + 61.15
Now, we calculate the value inside the sine function:
pi/3 - 2.09 ≈ 0.5236 - 2.09 ≈ -1.5664
Therefore:
y = 18.25 sin(-1.5664) + 61.15
Using a calculator to find the sine of -1.5664:
y ≈ 18.25 * (-0.9794) + 61.15
Simplifying further:
y ≈ -17.873 + 61.15
y ≈ 43.277
So, the average temperature for the month of February in Atlanta, GA is approximately 43.277°F.
Now, to find the average temperature for the month of June, we need to determine the value of "t" for June. Since June comes five months after January, the value of "t" for June will be 6.
Substituting "t = 6" into the given equation:
y = 18.25 sin(pi/6 * 6 - 2.09) + 61.15
Simplifying further:
y = 18.25 sin(pi/6 * 6 - 2.09) + 61.15
y = 18.25 sin(pi - 2.09) + 61.15
Note that sin(pi - x) = sin(x), so:
y = 18.25 sin(2.09) + 61.15
Using a calculator to find the sine of 2.09:
y ≈ 18.25 * 0.8910 + 61.15
Simplifying further:
y ≈ 16.017 + 61.15
y ≈ 77.167
Therefore, the average temperature for the month of June in Atlanta, GA is approximately 77.167°F.