Sinusoidal function

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 56 and 74 degrees during the day and the average daily temperature first occurs at 8 A.M. How many hours after midnight, to two decimal places, does the temperature first reach 61 degrees?

Answer 1

A= (74-56)/2= 18/2 = 9 = Amplitude

B=2pi/p=2pi/24= pi/12

C= phase shift

D = (74 +56)/2 =130/2= 65= midline

AsinB(t-C) +D

= 9sin[(pi/12)(8-C)+65 =65

sin[(pi/12)(8-C)]= 0

2pi/3 -piC/12 =0

C/12 =2/3

C= 8

f(t)= 9sin[(pi/12)(t-8)] +65 =61

sinx= -4/9

x= sin^-1(-4/9)= about -26.39 degrees = (pi/12(t-8) solve for t

-26.39pi/180= (pi/12)(t-8)

-26.39/15= t-8

t= 8am -1:46= 6:14 am