The sinusoidal function is represented as: y = A sin (ω(x - α)) + C
where: y = temperature in degrees; x = hours after midnight; A = amplitude; ω = 2π/period; α = Y-axis phase shift; and C = the midline.
In this problem, midline C = (38 + 52) / 2 = 45
Amplitude A = 52 - 45 = 7
Period = 24 hours
ω = 2π/24 = π/12
α = 10 (hours after midnight)
Therefore: y = A sin (ω(x - α)) + C
= 7 sin (π/12 (x - 10)) + 45
Solving this equation for y = 43, gives: 43 = 7 sin (π/12 (x - 10)) + 45
-2 = 7 sin (π/12 (x - 10))
-2/7 = sin (π/12 (x - 10))
(π/12 (x - 10)) = arcsin (-2/7)
And I leave solving for x to you, as an exercise.
The sinusoidal wave can be represented by the equation:y=A∗sin[ω(x−α)]+Cy=A∗sin[ω(x−α)]+C
where, A is the amplitude; ω=2π/periodω=2π/period; α=α= phase shift on the Y-axis; and C = midline.
With the information given in this problem,
Midline (C) is the average calculated as: (72+38)/2=55(72+38)/2=55;
Amplitude (A) is 72-55= 17;
Period = 24 hours;
Substituting in the equation,
Solving this equation for y=51y=51 gives the value of x as 9.09.
Thus, the temperature first reaches 51 degrees about 9.09 hours after midnight.