Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 52 degrees occurs at 3 PM and the average temperature for the day is 45 degrees.

Question

Find the temperature, to the nearest degree at 7 AM.

Answer 1

T(t) = Asin(Bt + C) + D; D = 45°, A = 52 - 45 = 7°, t is time in hours; B = 2π/24 = π/12

T(t) = 7sin(πt/12 + C) + 45.

At 3 PM time t = 15 hours (we count time from midnight) T = 52°,

52 = 7sin(15π/12 + C) + 45; sin(15π/12 + C) = 1; 5π/4 + C = π/2; C = - 3π/4

T(t) = 7sin(πt/12 - 3π/4) + 45.

At 7 AM t = 7 and T(7) = 7sin(7π/12 - 3π/4) + 45 = 7sin(-π/6) + 45 = 41.5° ≈ 42°