Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature of 64 degrees occurs at 4 PM and the average temperature for the day is 50 degrees. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

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.