Outside temperature over a day can be modeled as a sinusoidal function

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 33 and 57 degrees during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight, to two decimal places, does the temperature first reach 42 degrees?

Answer 1

Let's start by computing the average: (33 + 57) / 2 = 45

The amplitude of the sinusoid is 57 - 45 = 12. Thus the temperature is represented as:

T = 12 * sin(2π * (t + a) / 24) + 45.

T is temperature

t is time in hours after midnight

a is a "phase" term that allows us to set the time at which a particular temperature occurs.

We are told that T first reaches the average at 10 am. That means sin(2π * (10 + a)) = 0. That implies that 10 + a = 0, so a = -10.

Thus:

T = 12 * sin(2π * (t -10) / 24) + 45.

We want to know when T first hit 42 degrees. That means:

12 * sin(2π * (t -10) / 24) = -3

sin(2π * (t -10) / 24) = -1/4

Using a calculator, I find that 2π * (t -10) / 24 = -0.2526803. So all that remains is to solve for t:

t = -0.2526803 * 24 / 2π + 10

I get 9.034832 am. That's decimal time. The part after the decimal point can be converted to minutes and seconds. But I think this answers the question.