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.

Short 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-by-step solution

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 \[ 7.1 \cdot \lim_{t \to \infty} e^0 = 7.1 \cdot 1 = 7.1 \]

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 \[ V'(t) = 7.1 \cdot \frac{d}{dt}e^{(-48.1) / t} = 7.1 \cdot e^{(-48.1) / t} \cdot \frac{d}{dt}(-48.1) / t = 7.1 \cdot e^{(-48.1) / t} \cdot \frac{48.1}{t^2} \]

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 \[ V'(60) = 7.1 \cdot e^{(-48.1) / 60} \cdot \frac{48.1}{60^2} \]