Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 41 and 69 degrees during the day and the average daily temperature first occurs at 8 AM. How many hours after midnight, to two decimal places, does the temperature first reach 48 degrees?

(a). The sinusoidal function is

-14×cos((π/12)×(t-2))+55

(b). The temperature of 48 ° first occurs at 6.00 a.m.

Step-by-step explanation:

To solve the question, we note that an example of a sinusoidal function is a cosine function

a×cos×(2π/k)t+b = Temperature

For a 24 hour period, the sinusoidal function becomes

2π/k =24 or k = π/12 and

a = (69 - 41)/2 = 14 also b = 69 -14 = 55

Therefore the sinusoidal function becomes

14×cos((π/12)×t)+55 = Temperature at a particular time of day

checking we have

at 6 a.m. 14×cos(π/2) +55 = 55 ° okay

However the average temperature supposed to occur at 8 a.m.

Therefore we have time adjustment by 2 hours hence

Our equation becomes

-14×cos((π/12)×(t-2))+55 = Temperature

Therefore at 8 a.m. we have

-14×cos((π/12)×(8-2))+55 = 55 ° = average temperature

And

(b) For 48 °, we have

-14×cos((π/12)×(t-2))+55 = 48

or t = 6 a.m.

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