Question

The monthly sales (in thousands of units) of a seasonal product are approximated by

Answer 1

 Don't forget S is measured in thousands of units so you are solving for : 

100 < 74.5 + 43.75Sin(πt/6) 
25.5 < 43.75Sin(πt/6) 
Sin(πt/6) >25.5/43.75 = 0.582857 
ASrcSin(πt/6) > 0.62224 radians 
πt/6 > 0.62224 
t > 6 x 0.62224/π = 1.1884 (4dp) 

This initial value occurs when the sine value is increasing and it will reach its maximum value of 1 when Sin(πt/6) = Sinπ/2, that is when t = 3. 
Consequently, monthly sales exceed 100,000 during the period between t = 1.1884 and 4.8116 
[3 - 1.1884 = 1.8116 so the other extreme occurs at 3 + 1.8116] 

Note : on the basis of these calculations, January is 0 ≤ t < 1 : February is 1 ≤ t < 2 :....May is 4 ≤ t < 5 
So the period when sales exceed 100,000 occurs between Feb 6 and May 25 and annually thereafter.