Answer is: option2

Solution:

We are given the function:

S(t) = 0.46 cos(0.45t + 3.15) + 3.4

We are to find the time t when lumber sales were increasing at the greatest rate, i.e., when the derivative S'(t) is maximum.

Using the chain rule:

S'(t) = −0.46 · sin(0.45t + 3.15) · 0.45 = −0.207 sin(0.45t + 3.15)

We want maximum increase, i.e., maximum positive value of S'(t).

S'(t) = −0.207 sin(0.45t + 3.15)

This is maximum when sin(0.45t + 3.15) = −1, because of the negative coefficient.

So:
sin(0.45t + 3.15) = −1 ⟹ 0.45t + 3.15 = 3π/2 + 2πn, where n ∈ ℤ
0.45t = 3π/2 − 3.15 + 2πn

Using π ≈ 3.1416:

3π/2 ≈ 4.7124 ⟹ 0.45t ≈ 4.7124 − 3.15 = 1.5624 ⟹ t ≈ 1.5624 / 0.45 ≈ 3.472

Since t = 0 corresponds to 1980, t ≈ 3.47 years corresponds to:
1980 + 3.47 ≈ 1983.47

This falls within the year 1983.

Answer: B (1983)