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

S = 74.50 + 43.75 sin pi t/6

Where t is time in months, with t=1 corresponding to January. Determine the months when sales exceed 100,000 units

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.