Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 55 degrees at midnight and the high and low temperature during the day are 62 and 48 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.
the period is 24 hours, so if the minimum occurs at time h hours after 12:00, so start with something like
D = -cos(π/12 (t-h))
the amplitude is (62-48)/2 = 7
the central axis is (62+48)/2 = 55
aha! we see that D(0) is the minimum, so h=0.
we can revise our function to
D = 55 - 7cos(π/12 t)
there is no horizontal shift
To find an equation for the temperature, D, in terms of time, t, we can start by recognizing that a sinusoidal function can be written in the form:
D = A * sin(B * t + C) + D0
where:
- A is the amplitude, half the difference between the high and low temperatures,
- B is the frequency, related to the period of the function,
- C is the phase shift, indicating any horizontal translations,
- D0 is the vertical shift, the temperature at time t = 0.
To determine these values, we can use the given information:
- The high temperature is 62 degrees, and the low temperature is 48 degrees. The amplitude, A, is half the difference between these values: A = (62 - 48) / 2 = 7 degrees.
- The temperature at midnight is 55 degrees. Therefore, D0 is 55 degrees.
To determine B and C, we need to find the period of the function. The period is the length of time it takes for the temperature to complete one full cycle. In this case, it represents a full day, which is 24 hours.
Since the temperature is at its highest at some point during the day, the sinusoidal function reaches its maximum value B * t + C when t is the number of hours after the temperature reaches its maximum value. In this case, the maximum temperature is at noon, which is 12 hours after midnight (t = 12).
Using this information, we can determine C. Substituting t = 12 and D = 62 into the equation, we can solve for C:
62 = 7 * sin(B * 12 + C) + 55
Rearranging the equation, we get:
7 * sin(B * 12 + C) = 7
sin(B * 12 + C) = 1
Since the maximum value of sin(x) is 1, this implies that B * 12 + C must be in the form of (2 * n * π), where n is an integer. In other words, (B * 12 + C) is a multiple of 2 π.
We can choose the simplest case by setting C = 0, which means the sinusoidal function starts at its maximum value at t = 0. Therefore, B * 12 must be a multiple of 2 π.
Now, we can determine B. Dividing both sides of the equation B * 12 = (2 * π) by 12, we find B = (2 * π) / 12 = π / 6.
With these values, we can write the equation for the temperature, D, in terms of t:
D = 7 * sin((π/6) * t) + 55
Thus, the equation for the temperature, D, in terms of t is D = 7 * sin((π/6) * t) + 55.
To find an equation for the temperature, D, in terms of time, t, we can use a sinusoidal function in the form:
D = A*sin(B(t-C)) + D_avg
Where:
D is the temperature at time t,
A is the amplitude (half the difference between the high and low temperatures),
B is the frequency (2π divided by the number of hours it takes for the temperature to complete one full cycle),
C is the phase shift (the number of hours after midnight when the temperature is at its highest point),
D_avg is the average temperature.
Given:
Temperature at midnight = 55 degrees (D_avg),
High temperature = 62 degrees,
Low temperature = 48 degrees.
Average temperature:
D_avg = (High temperature + Low temperature) / 2
= (62 + 48) / 2
= 55 degrees
Amplitude:
A = (High temperature - Low temperature) / 2
= (62 - 48) / 2
= 7 degrees
Frequency:
The temperature goes through one cycle every 24 hours (one day). So, B = 2π/24 = π/12
Phase shift:
At the highest point, the temperature occurs 12 hours after midnight. So, C = 12.
Putting it all together, the equation for the temperature, D, in terms of time, t, is:
D = 7*sin((π/12)(t - 12)) + 55
Therefore, the equation for the temperature, D, in terms of t is D = 7*sin((π/12)(t - 12)) + 55.