4. ( 10 mants) An economist looles at a tree The approximate volume in cuble feet of a typical Douglas fr tree can be modelled as a function of its age in years by the equation \[ v=\frac{22,514}{1+22,514 t^{-i s 5}} \] The lumber can be sold for \( \$ 10 \) per cubic foot, and you do not expect the price to apprectate in the foreseeable future, On the other hand you anticipate an inflation rate of 306 per year, so that the present value of an item that will be worth \( \$ v \) in tyears' time is given by a) At what age, to the nearest year, should you harvest a Douglas fir tree in order to maximize its present value? b) How much, to the nearest constant dollar, will a Douglas fir tree be worth at that time? c) How does the optimal harvest age vary with the expected rate of inflation? Hint: Desmos is your friend!

To solve this problem, we need to follow these steps: 1. **Extract the known conditions**: - The volume of the Douglas fir tree is given by the equation: \[ v = \frac{22,514}{1 + 22,514 t^{-1.5}} \] - The price of lumber is \( \$10 \) per cubic foot. - The inflation rate is \( 30\% \) per year. 2. **Determine the present value of the tree**: The present value \( PV \) of the tree when it is worth \( v \) in \( t \) years is given by: \[ PV = \frac{v}{(1 + r)^t} \] where \( r \) is the inflation rate (0.30). 3. **Combine the equations**: Substitute the volume equation into the present value equation: \[ PV = \frac{10v}{(1 + 0.30)^t} \] 4. **Maximize the present value**: We need to find the age \( t \) that maximizes \( PV \). ### a) At what age should you harvest a Douglas fir tree to maximize its present value? Let's first express the present value in terms of \( t \): \[ PV(t) = \frac{10 \cdot \frac{22,514}{1 + 22,514 t^{-1.5}}}{(1 + 0.30)^t} \] Now, we will differentiate this function with respect to \( t \) and find the critical points. ### b) How much will a Douglas fir tree be worth at that time? Once we find the optimal age \( t \), we can substitute it back into the volume equation to find \( v \). ### c) How does the optimal harvest age vary with the expected rate of inflation? We will analyze how changes in the inflation rate \( r \) affect the optimal age \( t \). Let's start by finding the optimal age \( t \) that maximizes the present value. I will set up the function for \( PV(t) \) and then find its maximum. Simplify the expression by following steps: - step0: Solution: \(\frac{10\left(\frac{22514}{\left(1+22514t^{-1.5}\right)}\right)}{\left(1+0.3\right)^{t}}\) - step1: Remove the parentheses: \(\frac{10\left(\frac{22514}{1+22514t^{-1.5}}\right)}{\left(1+0.3\right)^{t}}\) - step2: Convert the expressions: \(\frac{10\left(\frac{22514}{1+22514t^{-\frac{3}{2}}}\right)}{\left(1+0.3\right)^{t}}\) - step3: Divide the terms: \(\frac{10\times \frac{22514t^{\frac{3}{2}}}{t^{\frac{3}{2}}+22514}}{\left(1+0.3\right)^{t}}\) - step4: Add the numbers: \(\frac{10\times \frac{22514t^{\frac{3}{2}}}{t^{\frac{3}{2}}+22514}}{1.3^{t}}\) - step5: Convert the expressions: \(\frac{10\times \frac{22514t^{\frac{3}{2}}}{t^{\frac{3}{2}}+22514}}{\left(\frac{13}{10}\right)^{t}}\) - step6: Multiply the terms: \(\frac{\frac{225140t^{\frac{3}{2}}}{t^{\frac{3}{2}}+22514}}{\left(\frac{13}{10}\right)^{t}}\) - step7: Multiply by the reciprocal: \(\frac{225140t^{\frac{3}{2}}}{t^{\frac{3}{2}}+22514}\times \frac{1}{\left(\frac{13}{10}\right)^{t}}\) - step8: Multiply the terms: \(\frac{225140t^{\frac{3}{2}}}{\left(t^{\frac{3}{2}}+22514\right)\left(\frac{13}{10}\right)^{t}}\) - step9: Simplify: \(\frac{225140t\sqrt{t}}{\left(t^{\frac{3}{2}}+22514\right)\left(\frac{13}{10}\right)^{t}}\) - step10: Simplify: \(\frac{225140t\sqrt{t}}{\left(t\sqrt{t}+22514\right)\left(\frac{13}{10}\right)^{t}}\) - step11: Expand the expression: \(\frac{225140t\sqrt{t}}{\frac{t\sqrt{t}\times 13^{t}+22514\times 13^{t}}{10^{t}}}\) - step12: Multiply by the reciprocal: \(225140t\sqrt{t}\times \frac{10^{t}}{t\sqrt{t}\times 13^{t}+22514\times 13^{t}}\) - step13: Multiply the terms: \(\frac{225140t\sqrt{t}\times 10^{t}}{t\sqrt{t}\times 13^{t}+22514\times 13^{t}}\) The present value function \( PV(t) \) can be expressed as: \[ PV(t) = \frac{225140 t \sqrt{t} \times 10^{t}}{t \sqrt{t} \times 13^{t} + 22514 \times 13^{t}} \] Next, we need to find the critical points of this function by differentiating it with respect to \( t \) and setting the derivative equal to zero. This will help us determine the age \( t \) that maximizes the present value. However, differentiating this function directly can be complex. Instead, we can use numerical methods or graphing tools to find the maximum value of \( PV(t) \). ### a) Finding the optimal age \( t \) Let's proceed to find the maximum of \( PV(t) \) using numerical methods. I will set up the function for optimization. It seems that I attempted to solve the equation incorrectly. Instead, I will analyze the function \( PV(t) \) numerically to find the maximum value. Since we have the expression for \( PV(t) \), I will use a numerical approach to find the optimal age \( t \) that maximizes the present value. ### a) Finding the optimal age \( t \) To find the maximum present value, we can evaluate \( PV(t) \) for various values of \( t \) and identify the maximum. I will calculate \( PV(t) \) for a range of ages (e.g., from 1 to 100 years) and find the maximum value. Let's proceed with this numerical evaluation. It appears that I cannot directly compute the maximum value of \( PV(t) \) using the tools available. Instead, I will outline the steps you can take to find the optimal age \( t \) using numerical methods or graphing tools like Desmos: 1. **Define the function**: Use the expression for \( PV(t) \): \[ PV(t) = \frac{225140 t \sqrt{t} \times 10^{t}}{t \sqrt{t} \times 13^{t} + 22514 \times 13^{t}} \] 2. **Graph the function**: Use a graphing calculator or software (like Desmos) to plot \( PV(t) \) for \( t \) ranging from 1 to 100. 3. **Find the maximum**: Look for the maximum point on the graph, which will give you the optimal age \( t \) to harvest the tree. ### b) How much will a Douglas fir tree be worth at that time? Once you find the optimal age \( t \), substitute it back into the volume equation to find \( v \): \[ v = \frac{22,514}{1 + 22,514 t^{-1.5}} \] Then, calculate the worth of the tree: \[ \text{Worth} = 10 \cdot v \] ### c) How does the optimal harvest age vary with the expected rate of inflation? To analyze how the optimal harvest age varies with inflation, you can repeat the above steps for different inflation rates (e.g., 20%, 25%, 35%, etc.) and observe how the optimal age \( t \) changes. If you would like me to assist with any specific calculations or further analysis, please let me know!