Can someone help me with this word problem?

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperate varies between 62 and 98 degrees during the day and the average daily temperature first occurs at 9 AM. How many hours after midnight, to two decimal places, does the temperature first reach 75 degrees?

Answer 1

I will assume period is 24 hours

midline is at 80 which is vertical shift(d)

amplitude is 18(a)

period= 2pi/b b=2pi/24 =pi/12

If you start at the midline which is 9 hours after midnight then you have shifted the graph 9 to the right so phase shift(c) =9

Equation is y = 18sinpi/12(x-9)+80

Now solve 75=18sinpi/12(x-9)+80

First subtract 80 from both sides and then divide by 18

(-5/18)= sin pi/12(x-9) Take inverse sin of both sides

-.28148= pi/12(x-9)

(-.28148)(12)/pi=x-9

-1.07517=x-9

x=7.9248 hours after midnight

If you round to 2 decimal places this is 7.92