Jeffrey K. answered • 10/02/20

Together, we build an iron base in mathematics and physics

Hi Rebecca:

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;

ω=2π/24ω=2π/24;

α=10α=10;

Substituting in the equation,

y=17∗sin[2π/24(x−10)]+55y=17∗sin[2π/24(x−10)]+55

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.