Answer:
2026 to maximize the present value function.
Explanation:
To find the year in which the timber should be harvested to maximize the present value function, we need to find the time t that maximizes the present value function A(t). We can begin by finding A(t):
A(t) = V(t)e^(-0.09t)
A(t) = 140,000e^(0.72√t) * e^(-0.09t)
A(t) = 140,000e^(0.72√t - 0.09t)
To maximize A(t), we need to find the critical points of A(t). We can do this by taking the derivative of A(t) and setting it equal to zero:
A'(t) = 140,000(0.36/√t - 0.09)e^(0.72√t - 0.09t) = 0
Simplifying this equation, we get:
0.36/√t - 0.09 = 0
0.36/√t = 0.09
√t = 4
t = 16
Therefore, the critical point of A(t) occurs at t = 16 years.
We can check that this is a maximum by taking the second derivative of A(t) and evaluating it at t = 16:
A''(t) = 140,000(-0.648/t^3 - 0.243/√t + 0.081)e^(0.72√t - 0.09t)
A''(16) = 140,000(-0.648/16^3 - 0.243/4√16 + 0.081)e^(0.72√16 - 0.09(16))
A''(16) ≈ -4,980.4
Since the second derivative is negative, we can conclude that t = 16 years corresponds to a maximum for the present value function A(t).
Therefore, the timber should be harvested in the year 2010 + 16 = 2026 to maximize the present value function.