Since it's sinusoidal, we're going to use the sin(x) function. All we have to do is make sure it has the right middle (smack dab between 78 and 92, instead of the usual 0), the right amplitude (half the difference between that 78 and 92), the right starting point and the right period.
The midline is (92+78)/2 = 85.
y = sin(x) + 85
would give us a sine function with midline at 85.
amplitude: (92-78)/2 = 14/2 = 7.
y = 7 sin(x) + 85
will give us the right amplitude and the right midline. Since it reaches its average (i.e. middle) temperature at 10 AM and the sine function starts at its midline going up, we shift the sine curve to be at that zero 10 hours after midnight:
y = 7 sin (x - 10) + 85
Finally, this function has to undergo its complete cycle every 24 hours. So, if we divide (x-10) by 24, then it will go from 0 to 1 in 24 hours. We want it to go from 0 to 2 Pi in that 24 hours instead, so we multiply by 2 Pi, giving us our function:
y = 7 sin((x-10) * (2 Pi)/24) + 85
simplifying to:
y = 7 sin( Pi * (x-10) /12 ) + 85
where
y = temperature in degrees
x = number of hours since midnight.
To answer the specific question given, plug in 83 for y and then solve for x.
