Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 70 degrees at midnight and the high and low temperature during the day are 76 and 64 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.
Question
Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 70 degrees at midnight and the high and low temperature during the day are 76 and 64 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.
Solution
To model the temperature as a sinusoidal function, we can use the sine or cosine function. The general form of a sinusoidal function is:
D(t) = A sin(B(t - C)) + D
where:
- A is the amplitude (half the difference between the maximum and minimum values),
- B determines the period (the length of one complete cycle),
- C is the horizontal shift (how the graph is shifted horizontally), and
- D is the vertical shift (the average value).
Given that the high and low temperatures during the day are 76 and 64 degrees respectively, we can find the amplitude (A) and the vertical shift (D) as follows:
A = (Max - Min) / 2 = (76 - 64) / 2 = 6 degrees D = (Max + Min) / 2 = (76 + 64) / 2 = 70 degrees
The period of the temperature function is 24 hours because it repeats every 24 hours. So, B = 2π / Period = 2π / 24.
The temperature is at its average value at midnight and it increases to the maximum at noon. Therefore, the function is shifted to the right by 6 hours. So, C = 6.
Substituting A, B, C, and D into the general form gives us the temperature function:
D(t) = 6 sin((2π / 24) * (t - 6)) + 70
This function gives the temperature, D, in terms of the number of hours since midnight, t.
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