Modeling with Trigonometric Functions

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 26 and 64 degrees during the day and the average daily temperature first occurs at 9 AM. How many hours after midnight, to two decimal places, does the temperature first reach 36 degrees?

Answer 1

T = Asin(Bt + C) + D;'

A = (64 - 26)/2 = 19 and D = (64 + 26)/2 = 45. So, T = 19sin(Bt + C) + 45.

B = 2π/24 = π/12.

Now T = 19sin(πt/12 + C ) + 45. When t = 9 T = 45.

So, 45 = 19sin(9π/12 + C) + 45; 19sin(3π/4 + C) = 0; 3π/4 + C = π; C = π/4.

At last T = 19sin(πt/12 + π/4) + 45;

36 = 19sin(πt/12 + π/4) + 45; sin(πt/12 + π/4)= - 9/19; πt/12 + π/4 = π - sin^-1(- 9/19) = 3.635062;

t = 12(3.635062 /π - 1/4) = 10.885 hours