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.