Problem 93 - Essential Calculus

Question

Timber Yield The yield \(V\) (in millions of cubic feet per acre) for a stand of timber at age \(t\) (in years) is \(V=7.1 e^{(-48.1) / t}\) (a) Find the limiting volume of wood per acre as \(t\) approaches infinity. (b) Find the rates at which the yield is changing when \(t=20\) years and \(t=60\) years.

Step-by-Step Solution

Answer

The limiting volume of wood per acre as \(t\) approaches infinity is approximately 7.1 million cubic feet. The rates of change of the yield when \(t=20\) and \(t=60\) years can be found by substituting these values into the derivative that was calculated.

Step 1: Find the Limit as t approaches Infinity

To find the limiting volume of wood as \(t\) approaches infinity, we can apply the concept of limits. In this scenario, it is required to find the limit of the function \(V=7.1 e^{(-48.1) / t}\) as \(t\) approaches infinity. \n \[ \lim_{t \to \infty} 7.1 e^{(-48.1) / t} = 7.1 \cdot \lim_{t \to \infty} e^{ - 48.1 / t } \] \n Since any number to the power -t as t approaches infinity is \(0\), we have \n \[ 7.1 \cdot \lim_{t \to \infty} e^0 = 7.1 \cdot 1 = 7.1 \]

Step 2: Differentiate the Function

Next, in order to find the rate at which the yield us changing, we must find the derivative of the function \(V(t)\). The derivative of the function \(V(t)=7.1 e^{(-48.1) / t}\) can be found using the chain rule and the derivative of constant \(e\). The derivative is: \n \[ V'(t) = 7.1 \cdot \frac{d}{dt}e^{(-48.1) / t} = 7.1 \cdot e^{(-48.1) / t} \cdot \frac{d}{dt}(-48.1) / t = 7.1 \cdot e^{(-48.1) / t} \cdot \frac{48.1}{t^2} \]

Step 3: Evaluate Derivative at Specific Points

Now to find the rates of change at \(t=20\) and \(t=60\) years, we substitute these values into the derivative. The rates of change at \(t=20\) and \(t=60\) years are: \n \[ V'(20) = 7.1 \cdot e^{(-48.1) / 20} \cdot \frac{48.1}{20^2} \] and \n \[ V'(60) = 7.1 \cdot e^{(-48.1) / 60} \cdot \frac{48.1}{60^2} \]

Key Concepts

Exponential Functions

Exponential functions are a key concept in calculus and other areas of mathematics and sciences. They are highly useful for modeling situations where things grow or decay at rates proportional to their current size, such as populations, investments, or even timber yields.
An exponential function is typically expressed as \(f(t) = a \cdot e^{b \cdot t}\), where \(a\) is a constant multiplier, \(e\) is the base of the natural logarithms (approximately 2.71828), and \(b\) is the growth rate. In the given problem, the function \(V(t) = 7.1 e^{(-48.1) / t}\) models the yield from a stand of timber as it ages. Here:

• \(7.1\) represents the maximum yield in millions of cubic feet per acre, effectively scaling the function.
• The exponent \((-48.1 / t)\) signifies how the yield changes as the timber ages, capturing the decay rate over time as \(t\) increases.

Learning to work with exponential functions is crucial for understanding natural growth and decay processes. Recognizing their presence in equations allows students to model and solve real-world problems.

Derivative Calculations

Calculating derivatives is an essential part of calculus as it helps determine the rate of change of a function. When working with exponential functions like \(V(t) = 7.1 e^{(-48.1) / t}\), we use the chain rule to handle derivatives involving products of functions.
The chain rule is used when differentiating a composite function, such as \(e^{u(t)}\). It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function itself.

• For the exponential part, \(\frac{d}{dt}e^{(-48.1) / t} = e^{(-48.1) / t} \times \frac{48.1}{t^2}\), obtained by differentiating \((-48.1 / t)\).
• We then multiply by 7.1, our scaling constant, to account for the initial maximum yield.

This derivative tells us the sensitivity or responsiveness of the yield with respect to changes in time, allowing us to predict future behavior of the vegetation growth in the forest or manage resources effectively.

Rates of Change

Understanding rates of change in the context of calculus involves examining how a function transforms over time. The rate of change is represented by the derivative of the function. For the timber yield example, we use the derivative to find out how rapidly the yield is increasing or decreasing at specific times.
Let's delve into calculating the rate of change of the function \(V(t)\) at particular ages:

• At \(t=20\) years, substituting into the derivative \(V'(t) = 7.1 e^{(-48.1) / t} \frac{48.1}{t^2}\) allows us to find \(V'(20)\), showing how the yield changes at this age.
• Similarly, at \(t=60\) years, substituting gives us \(V'(60)\), revealing the rate of change as the forest matures.

These calculations provide insights into the effect age has on timber yields. Understanding these rates helps in making informed decisions regarding harvesting schedules and sustainable practices, which are critical for forestry management. By predicting how quickly or slowly resources appear, better strategies can be devised to ensure optimal and sustainable use of natural resources.