Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 94 degrees occurs at 6 PM and the average temperature for the day is 75 degrees. Find the temperature, to the nearest degree, at 7 AM.
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To find the temperature at 7 AM, we need to determine the equation of the sinusoidal function.
Let's assume that the temperature can be modeled by the equation:
T(t) = A * sin(Bt + C) + D
Where:
T(t) is the temperature at time t,
A is the amplitude of the sinusoidal function,
B is the period of the function,
C is the phase shift,
D is the vertical shift.
From the given information, we know that the high temperature of 94 degrees occurs at 6 PM. In 24-hour format, 6 PM is equivalent to 18:00. We can use this information to determine the phase shift (C).
C = (18/24) * 2π
C = 3π/4
Since sinusoidal functions have a period of 2π, the period (B) is 2π.
Given that the average temperature for the day is 75 degrees, we can use this information to determine the vertical shift (D).
D = 75
The amplitude (A) is half the difference between the high temperature and the average temperature.
A = (94 - 75) / 2
A = 19 / 2
A = 9.5
Thus, the equation for the temperature over the day is:
T(t) = 9.5 * sin(2πt + 3π/4) + 75
To find the temperature at 7 AM, we can substitute t = 7/24 into the equation:
T(7/24) = 9.5 * sin(2π * 7/24 + 3π/4) + 75
T(7/24) ≈ 9.5 * sin(7π/12 + 3π/4) + 75
T(7/24) ≈ 9.5 * sin(19π/12) + 75
Now, we can calculate the temperature by substituting the value into a calculator or using a table of values for the sine function. Rounding the result to the nearest degree, we get:
T(7/24) ≈ 9.5 * (-0.707) + 75
T(7/24) ≈ -6.71 + 75
T(7/24) ≈ 68
Therefore, the temperature at 7 AM is approximately 68 degrees.
To find the temperature at 7 AM, we need to determine the equation that models the temperature variation over the day as a sinusoidal function.
Given that the high temperature occurs at 6 PM (which is 12 hours after 6 AM) and the average temperature for the day is 75 degrees, we can start by assuming the average temperature is also the midline of the sinusoidal function.
The general form of a sinusoidal function is:
f(x) = A*sin(B(x - C)) + D
where:
A represents the amplitude (half the difference between the maximum and minimum values)
B represents the number of cycles (2π/B) that the function completes within a given interval
C represents the phase shift (horizontal shift)
D represents the vertical shift (midline)
In this case, the midline is 75 degrees, and the amplitude can be calculated as half the difference between the high and average temperature:
Amplitude = (Maximum temperature - Average temperature) / 2 = (94 - 75) / 2 = 9.5 degrees
Since the high temperature occurs at 6 PM, which is 12 hours later than 6 AM, we have a phase shift of 12 hours. In terms of radians, this is equivalent to 2π.
Finally, we can write the equation for the temperature variation as:
f(x) = 9.5*sin((2π/24)*(x - 12)) + 75
To find the temperature at 7 AM, we substitute x = 7 into the equation:
f(7) = 9.5*sin((2π/24)*(7 - 12)) + 75
≈ 9.5*sin(-π/4) + 75
≈ 9.5*(-0.707) + 75
≈ -6.72 + 75
≈ 68.28
Rounded to the nearest degree, the temperature at 7 AM is approximately 68 degrees.
Since we know when the max occurs, use a shifted cosine, since cos(0) is the max
Since the period is 24, cos(kx) has period 2π/k
y = 75 + (94-75)cos(π/12 (x-18))
so find y(7)