Question 1179528
The sinusoidal variation with temperature will have the general form:
T(t) = T0 + Asin(wt - phi), where T0 is the average temperature, A is
the amplitude of variation, w is the angular frequency, and phi is a phase shift
Since the day starts at midnight, i.e. t = 0, and the day does not
necessarily start out at the average temperature, we need to introduce
a phase shift. We are told that the temperature varies between 60 and 80,
thus the average temperature = 70, and the amplitude of variation about the
average is 10. Since the temperature completes one cycle in 24 h, the
angular frequency w = 2*pi/24 = pi/12
Since the average temperature is first reached at 8:00 am, which is 8 hours
after midnight, sin(pi*8/12 - phi) = 0 -> phi = 8/12*pi = (2/3)pi
Thus, the equation is T(t) = 70 + 10sin((pi/12)t - (2pi/3))
T(t) = 65 = 70 + 10sin((pi/12)t - (2pi/3)) -> -1/2 = sin((pi/12)t - (2pi/3))
Since asin(-1/2) = -pi/6 -> (pi/12)t - 2pi/3 = -pi/6
Solving for t gives t = 6. Thus the temperature is 65 at 6:00 am