Assuming t is the number of hours since midnight, find an equation for the temperature, T, in terms of t.

Question

The temperature during the day can be modeled by a sinusoid. Answer the following question given that the low temperature of 17 degrees occurs at 3 AM and the high temperature for the day is 45 degrees.



Assuming t is the number of hours since midnight, find an equation for the temperature, T, in terms of t.

Answer 1

So sinusoid basically means that it has a sine function. So if the highest and lowest temperatures are 45 and 17 degrees, then the mid line of the function will be halfway in between those, or at 31 degrees. The amplitude would then be 45-31=14 degrees. Now we know that the low temperature occurs at 3 AM, and because this is sinusoidal, then the high temperature must be at 3 PM; this also means the period is 24 hours.

Now let's translate that into the equation.

T = 14sin(at+b) + 31 {This is the function y=Asin(θ+b) + c}

Then then to find the a and b:

24a = 2π, so a=π/12 {This is because the 24 hour period should equal a full period on the unit circle: 2π}

3a+b = π/4+b = -π/2, so b = -3π/4 {The -π/2 is a low point on the standard sine graph}

This makes the final equation as: T = 14sin(^πt/₁₂ - ^3π/₄) + 31.