The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the

temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively.

Assuming t is the number of hours since midnight, nd a function for the temperature, D, in terms of t.

D = - 12sin (π/12) (t) + 68

Step-by-step explanation:

Let temperature D in terms of number of hours t be given as:

D = asinkt + 68

Now,

The difference between high and low temperatures is = 80-56 = 24 °F

The period is = 24 hours

So, we have 2π/k = 24

Or, k = 24/2π

Now,

Let a = 12 hours

So, our equation becomes

D = 12sin (2π/24) (t) + 68

This is valid for 12 hours gap. If we want to implement for whole day then

D = - 12sin (π/12) (t) + 68

Put t=0

D = 68°F which is temperature at midnight

Put t=6

D = - 12 (1) + 68

D = 56°F

Put t=18

D = - 12 (-1) + 68

D = 80°F