Outside temperature over a day can be modeled as a sinusoidal function.

Question

Suppose you know the high temperature of 67 degrees occurs at 5 PM and the average temperature for the day is 50 degrees. Find the temperature, to the nearest degree, at 7 AM.

Answer 1

We are looking for an equation which looks like: T=Asin(Bt+C)+D where T is the temperature and t is time of the day.

We need to find A, B, C and D

Since this is modeled over a day period we have period is 24 therefore P=2pi/B so B=pi/12

D = 50 which is average of the day

Amplitude A is half a distance from average to maximum therefore A=17

So far we have:

T=17sin(pi/12t+C)+50

To find C we use point 5 pm when the temperature is 67 or(17,67)

67=17sin(pi/12 * 17 +C)+50

17=17sin(pi/12 * 17 +C)

1=sin(pi/12 * 17 +C)

17pi/12+C=pi/2

C=pi/2-17pi/12

C=6pi/12-17pi/12

C=-11pi/12

Final equation is

T=17sin(pi/12 t -11pi/12)+50

To find temperature at 5 am just plug in t=5

T=17sin(pi/12 * 5 -11pi/12)+50=49.53