Problem 107 - College Algebra and Calculus: An Applied Approach

Question

The yield \(V\) (in millions of cubic feet per acre) for a forest at age \(t\) years is given by \(V=6.7 e^{-48.1 / t}, \quad t>0\) (a) Use a graphing utility to find the time necessary to obtain a yield of \(1.3\) million cubic feet per acre. (b) Use a graphing utility to find the time necessary to obtain a yield of 2 million cubic feet per acre.

Step-by-Step Solution

Answer

The answer will depend on the solution provided by the graphing utility used to compute \(t\). The obtained \(t\) values will be the required time to achieve the mentioned yield amounts.

Step 1: Solve for time to obtain a yield of 1.3 million cubic feet

To solve this, put \(V = 1.3\) in the given formula and solve for \(t\).\nHence, formula becomes, \(V=6.7 e^{-48.1 / t}, \quad t>0\).\nIn this problem, instead of plugging values yourself, a graphing utility can be used to find \(t\) when \(V = 1.3 million\).

Step 2: Solve for time to obtain a yield of 2 million cubic feet

Similar to previous step, put \(V = 2\) in the yield formula and solve for \(t\).\nSo the formula to solve becomes, \(V=6.7 e^{-48.1 / t}, \quad t>0\)\nJust like the previous step, in this scenario, we will plug in the value of \(V\) into a graphing calculator and solve for \(t\) when \(V = 2 million\) cubic feet

Key Concepts

Graphing Utility

A graphing utility is a handy tool that allows you to visualize and interact with mathematical functions. Instead of manually calculating complex equations, you can rely on technology to streamline the process. Here's how a graphing calculator or software can help manage exponential functions:

• Plot Complex Functions Easily: The ability to plot exponential functions, like the forest yield function provided, helps you instantly see how the yield changes with time.
• Accurate Calculations: Entering the complete formula into a graphing utility reduces the risk of errors that may occur in manual calculations, ensuring precision.
• Interactive Tools: Some utilities allow you to change parameters or zoom in on graphs, offering a deeper understanding of function behavior over various ranges.
• Finding Specific Values: As seen in the exercise, adjusting parameters to find out the exact age of a forest to reach given yields involves solving for unknowns like 't' in our function. Graphing utilities make this task manageable by providing precise values through visual markers or calculated points.

In the context of our yield problem from a forest, using a graphing utility transforms what could be a time-consuming calculation into a swift and clear solution. By setting values for yields and letting the utility compute, students can efficiently find the precise ages when the forest reaches specific yields.

Yield Calculation

Yield calculation in forestry often involves understanding the relationship between forest age and the volume of harvestable wood. Exponential functions, like the one given in this problem, help model these complex relationships. The forest yield formula provided is:\[ V=6.7 e^{-48.1 / t}, \quad t>0 \] This equation shows how forest yield (\( V \)) changes with forest age (\( t \)). Here's a closer look at the calculation:

• Equation Components: The number 6.7 represents the maximum potential yield in millions of cubic feet per acre. The term \( e^{-48.1/t} \) indicates how rapidly the yield approaches this value as the forest ages.
• Solving for Specific Yields: By setting \( V \) to a specific value, as we did for 1.3 and 2 million cubic feet, you can find the corresponding time \( t \). Exponential decay in this context means calculating exactly when the forest reaches a particular maturity level, which is facilitated through a graphing utility or algebraic manipulation.
• Applications: Understanding these calculations helps plan sustainable logging practices, forecast economic return from timber sales, and estimate long-term forest management strategies.

Yield calculations built on exponential function models are not just academically stimulating; they also play a pivotal role in realistic forest management scenarios.

Forest Yield Modeling

Forest yield modeling involves using mathematical functions to predict how forests grow and produce valuable resources over time. This area of study helps foresters make informed decisions on forest management. Let's delve into the primary elements:

• Understanding Exponential Growth: In our scenario, the equation \( V=6.7 e^{-48.1 / t} \) typifies how growth rate decreases over time. Initially, the growth is fast, but it slows as the forest becomes older. This decay aligns with ecological studies that young forests typically grow faster.
• Sustainable Management: Modeling yields enables foresters to devise sustainable harvesting cycles, balancing ecological health and economic benefit. By projecting future growth, managers can schedule logging activities to minimize ecological disruption.
• Practical Implications: Real-world applications include estimating when to conduct thinning activities, planning future cutting, and assessing forest health trends. Managers can use insights from yield models in making policy decisions and setting conservation priorities.
• Technology Integration: Modern models often integrate with advanced data analytics and satellite imagery, enhancing prediction accuracy and forest resource valuation. Using models coupled with real data feedback creates a robust framework for effective forest management.

Modeling forest yields is a powerful tool that not only supports academic learning but also underpins vital management strategies for sustainable forest use and conservation. By mastering these concepts, students gain valuable insights into the interaction between ecological processes and resource management.