How many hours after midnight does the temperature first reach 81 degrees?

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 70 and 100 degrees during the day and the average daily temperature first occurs at 9 AM.

Answer 1

What a fun word, "sinusoidal." Just means a modified sine function, which we'll build.

Temp = A sin ( B (x - C) ) + D

Temperature varies between 70 and 100, so the average is D=85 and the range is ±15=A.

360° / 24 hours = 15 = B

Average is at 9am, which is the start of our function, so C=9.

Thus our sinusoidal function is Temp = 15 sin (15 (x-9) ) + 85 where x is time, starting at 9 (9am).

Set Temp=81 and solve for x. You can do that part yourself. You can even use the negative angle to count the hours back from 9am.