Trigonometry homework

Question

Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature for the day is 71 degrees and the low temperature of 49 degrees occurs at 4 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

Answer 1

D(t)= AsinB(t-C)+E

A=amplitude=(71-49)/2=22/2=11

B=2pi/period=2pi/24= pi/12

C= phase shift=-14

E=midline=average temperature=(71+49)/2=120/2=60

D(t)=11sin[(pi/12)(t-C)]+60

t=hours since midnight, 4am is t=4

D(4)=49=60-11

sin[pi/12(4-C)]=-1

(pi/12)(4-C)=3pi/2

(4-C)/6=3

4-C=18

C=-14

D(t)=11sin[(pi/12)(t+14)]+60 is the sinusoidal function with low 49 at 4am, high 71 at 4pm

D(4)=11sin[pi/12)(4+14)]+60

=11sin(18pi/12)+60

=11(-1)+60

=49 at 4am