how do I solve?

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 60 degrees at midnight and the high and low temperatures during the day are 74 and 46 degrees, respectively. Assuming xx is the number of hours since midnight, find an equation that models the temperature over the course of the day.

y=   

Answer 1

consider shape of temperature flux:

low around 6 am; high 6 pm

repeats every 24 hours

this shape looks like the negative sine function: y= -sin(x)

the x value is in hours so the "inside" sine will be either (x(2π/24)) radians or (x(360/24)) degrees

( notice its 1 cycle/independent units) 1 cycle is 1 day which is 24 hours. 1 cycle is either 2π or 360°

there is no x offset as T(0)=60 which is the midline of this sin function.

high-low divide by 2 is the amplitude (74-46)/2 = +14 degrees ampliude: difference /2

the midline is (74+46)/2 = +60 degrees midline: sum/2

building it: (in radians)

y=-sin(x)

y=-sin(x(2π/24))

y=-14sin(xπ/12)

y=-14sin(xπ/12)+60

test: T(0)=60 √√ T(18)=14+60=74 degrees √√

building it in degrees:

y=-sin(x)

y=-sin(x(360/24))

y=-14sin(x(360/24))

y=-14sin(15x)+60

test: T(0)=60 √√ T(18)=14+60=74 degrees √√

Answer 2