help its due in a hour precal

Question

Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature for the day is 100 degrees and the low temperature of 70 degrees occurs at 3 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

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Answer 1

General sine function a sin ( b (t-c)) + d

High is 100

Low is 70

a = (100-70) / 2 = 15

d = ( 100 + 70) / 2 = 85

Assuming the high and low temperatures occur every 24 hours, then period = 24

b= 2π / period = π/12

The lowest point of a sine function occurs 3/4 way through the period ie after 18 hours

So we need that when t=3 not when t = 18, so we need to shift the sine function left 15 units, so c will be + 15

So the equation is 15 sin ( (π/12) (t + 15)) + 85

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You can draw this on Desmos and check it fits (I just did that)