Precalc Chapter 6.1

Question

Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature for the day is 58 degrees and the low temperature of 42 degrees occurs at 5 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

Answer 1

A sinusoid is either a sine or cosine wave considering the following properties:

D=a•sin(b(t-c))+d

1. a = Amplitude
2. 2𝛑/b = Period
3. c = phase (horizontal) shift
4. d = vertical shift

Where

D = temperature

t = time of the day (within 24 hour period) where 0 ≤ t < 24

For example if t = 5.5 is 5:30 AM and t=13 is 1:00PM as stated in the problem.

This problem will only make sense if one period is within 24 hours (one day).

Amplitude = a = (max-min)/2 = (58-42)/2 = 8

Period = 2𝛑/b = 24

2𝛑/24 = b

b = 𝛑/12

In this case, the lowest temperature is -8 degrees. To change it to 42, we need to get the vertical shift:

-8 + d = 42

d = 50

Remember that in the sine function, the highest point is at 1/4 of a period and the lowest is at 3/4 of a period starting from the origin and we want it to be equal to 5 AM.

5 = (3/4) (24) + c

5 = 18 + c

c = -13

Therefore, the sinusoidal function for this is:

D = 8•sin [(𝛑/12)(x - (-13))] + 50

D = 8•sin [(𝛑/12)(x +13)] + 50