Let's write a function of the form M(t) = Asin(B(t - C)) + D where t: time after midnight (hrs) ; M(t): outside temp
What we need to do is use the given info to find the valkues for the 4 parameters above: A, B, C, and D.
Let's start with D:
D represents the y-value (temp) of the sinusoidal axis (average temp) and we calculate it by averaging the highest and lowest temps reached. So, D = (43 + 77) / 2 = 60.
Next, we find A, the amplitude, by taking the difference of those hi and low temps divided by 2: A = (77-43)/2 = 17. Happily, this is also how much more the hi temp is than the avg, and how much less the low temp is than the avg. (It is useful to think of this sinusoid as one that goes 17 above 60, and 17 below 60.)
Next, we can calculate B, which is determined by the period of the sinusoid (i.e. the length of time for the curve to complete one full cycle), and is found by B = 2π / period. Thus, B = 2π / 24 = π / 12.
Lastly, since the temp reaches an average value at 8 am (t = 8) and is rising at that time, we should shift the function to the right 8 (think graph transformations) by letting C = 8.
M(t) = 17sin(π/12(t - 8)) + 60
Finally, set M(t) = 48 and solve, using inverse sine at some point to solve for t. Estimating gives me an answer very close to 5 am.
