Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 95 degrees occurs at 3 PM and the average temperature for the day is 80 degrees. Find the temperature, to the nearest degree, at 8 AM
amplitude is (95-80)=15
center line is 80, so start with the assumption that the minimum occurs at midnight (0 hours):
y = -15cos(kx)+80
The function has a period of 24 hours, so 2?/k = 24, making k=?/12
y = -15cos(?/12 x) + 80
Now we know that the max occurs at x=15, not x=12, so we need to shift by 3:
y = -15cos(?/12 (x-3)) + 80
See the graph at
http://www.wolframalpha.com/input/?i=plot+-15cos(%CF%80%2F12+(x-3))+%2B+80++for+0%3C%3Dx%3C%3D24
To find the temperature at 8 AM, we need to determine the equation of the sinusoidal function that models the temperature throughout the day.
Let's denote the time in hours as 't' and the temperature as 'T'.
Given that the high temperature of 95 degrees occurs at 3 PM, we can determine the phase shift of the sinusoidal function. Since 3 PM is 15 hours after midnight, the phase shift is 15.
The average temperature for the day is 80 degrees, which represents the midline of the sinusoidal function.
Therefore, the equation for the sinusoidal function is:
T = A * sin(B(t - C)) + D
Where:
- A is the amplitude,
- B is the period, which is 2π divided by the duration of one period in hours,
- C is the phase shift, and
- D is the midline (average temperature).
We know D = 80 and C = 15.
To find A and B, we need to determine the period. Since the high temperature occurs at 3 PM and the low temperature should occur 12 hours later, at 3 AM the next day, the period is 24 hours. Therefore, B = 2π/24 = π/12.
To find A, we need to determine the amplitude. The amplitude is half the difference between the highest and lowest temperatures. From the given information, the highest temperature is 95 degrees and we want to find the temperature at 8 AM, which is 5 hours after midnight. We can write this as T(8) = A * sin(B(8 - 15)) + 80. Simplifying this equation, we get:
95 = A * sin((-7π)/12) + 80
Subtracting 80 from both sides, we have:
15 = A * sin((-7π)/12)
Dividing by sin((-7π)/12), we get:
A = 15 / sin((-7π)/12)
Using a calculator, we find A ≈ 18.72.
Now we can calculate the temperature at 8 AM:
T(8) = A * sin(B(8 - 15)) + 80
T(8) = 18.72 * sin((π/12)(-7)) + 80
T(8) ≈ 18.72 * sin(-7π/12) + 80
T(8) ≈ 18.72 * (-0.866) + 80
T(8) ≈ -15.96 + 80
T(8) ≈ 64.04
Therefore, to the nearest degree, the temperature at 8 AM is 64 degrees.
To find the temperature at 8 AM, we can use the given information and model the temperature as a sinusoidal function.
Let's start by defining the function. A sinusoidal function can be represented as:
T(t) = A * sin(B(t - C)) + D
Where:
T(t) is the temperature at time t
A is the amplitude (half the difference between the highest and lowest temperatures)
B is the frequency (2π divided by the period in hours)
C is the phase shift (the time at which the sinusoidal function reaches its maximum value)
D is the vertical shift (the average temperature for the day)
From the given information, we have:
The high temperature of 95 degrees occurs at 3 PM, which is 15:00.
Let's consider this as the maximum value of the sinusoidal function, so C = 15.
The average temperature for the day is 80 degrees, which is the vertical shift, so D = 80.
To find the amplitude and frequency, we can use the fact that a full period of the sinusoidal function occurs over 24 hours.
Since the high temperature occurs at 15:00 and the full period is 24 hours, the low temperature should occur 12 hours before or after the high temperature. Therefore, the low temperature should occur at 3 AM or 3 PM the next day.
Since we need to find the temperature at 8 AM, which is 5 hours after the low temperature, we can consider this as a quarter of a full period.
Knowing this, we can set up the equation:
Temperature at 3 AM/PM + amplitude = Temperature at 8 AM
T(3) + A = T(8)
Given that T(3) = 80 (the average temperature) and T(3) + A = 95 (the high temperature), we can solve for the amplitude (A):
80 + A = 95
A = 95 - 80
A = 15
Now that we have the amplitude, we can calculate the frequency. A full period occurs over 24 hours, so the frequency is 2π divided by 24:
B = 2π / 24
B ≈ 0.2618
Finally, we can calculate the temperature at 8 AM by substituting the values into the sinusoidal function:
T(8) = 15 * sin(0.2618 * (8 - 15)) + 80
Simplifying further:
T(8) = 15 * sin(0.2618 * (-7)) + 80
T(8) ≈ 15 * sin(-1.8326) + 80
T(8) ≈ 15 * (-0.9892) + 80
T(8) ≈ -14.838 + 80
T(8) ≈ 65.162
Therefore, the temperature at 8 AM is approximately 65 degrees Fahrenheit.