The following equation models the approximate volume in...

The following equation models the approximate volume in cubic feet of a typical Douglas fir tree of age $t$ years. ${ }^{25}$ $$V=\frac{22,514}{1+22,514 t^{-2.55}}$$ The lumber will be sold at $$\$ 10$$ per cubic foot, and you do not expect the price of lumber to appreciate in the foreseeable future. On the other hand, you anticipate a general inflation rate of $$5 \%$$ per year, so that the present value of an item that will be worth in t years' time is given by $$p=v(1.05)^{-t}$$ At what age (to the nearest year) should you harvest a Douglas fir tree in order to maximize its present value? How much (to the nearest constant dollar) will a Douglas fir tree be worth at that time?

The following equation models the approximate volume in cubic feet of a typical Douglas fir tree of age $t$ years. ${ }^{25}$
$$V=\frac{22,514}{1+22,514 t^{-2.55}}$$
The lumber will be sold at $$\$ 10$$ per cubic foot, and you do not expect the price of lumber to appreciate in the foreseeable future. On the other hand, you anticipate a general inflation rate of $$5 \%$$ per year, so that the present value of an item that will be worth in t years' time is given by
$$p=v(1.05)^{-t}$$
At what age (to the nearest year) should you harvest a Douglas fir tree in order to maximize its present value? How much (to the nearest constant dollar) will a Douglas fir tree be worth at that time?

Key Concepts

Optimization using Calculus

This concept involves finding the maximum or minimum value of a function by taking its derivative with respect to the independent variable, setting the derivative equal to zero to identify critical points, and then analyzing these points to determine which one corresponds to the optimum. In the context of the problem, calculus is used to determine the age at which the present value of the tree’s lumber is maximized.

Time Value of Money and Present Value

The time value of money is a financial principle stating that a dollar today is worth more than a dollar received in the future due to its earning potential. Present value calculations incorporate this by discounting future cash flows back to their value today using a discount factor or rate (in this case, an inflation rate), allowing for a fair comparison between values received at different times.

Exponential Discounting

Exponential discounting is the method of reducing the value of future cash flows by a constant percentage per time period, represented mathematically as an exponential function. This concept is crucial for converting a future value into a present value, accounting for inflation or other forms of devaluation over time.

Mathematical Growth Modeling

Growth modeling uses mathematical functions to simulate and predict the development of real-world processes over time. In this scenario, the volume of a tree is modeled as a function of its age, allowing for the representation of how its physical characteristics change as it grows, which is essential for determining the optimal time to harvest.

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