Chapter 3.4, Problem 150E

a.

To determine

To use : the graphing utility to graph the function

Explanation of Solution

Given information : The question is

The yield for a forest at age $t$ is given by $V=6.7e^{\raisebox{1ex}{$-48.1$}\!\left/ \!\raisebox{-1ex}{$t$}\right.}$

Graph : Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $6.7e^{\raisebox{1ex}{$-48.1$}\!\left/ \!\raisebox{-1ex}{$t$}\right.}$ which is required to graph.

Step 4: Press $\boxed{\text{GRAPH}}$ button to graph the function.

The graph is obtained as:

  

Interpretation :

From the above graph the red line shows the function $6.7e^{\raisebox{1ex}{$-48.1$}\!\left/ \!\raisebox{-1ex}{$t$}\right.}$

b.

To determine

To calculate : the horizontal asymptote of the function

Answer to Problem 150E

The horizontal asymptote is x-axis component to $\infty$

Explanation of Solution

Given information : The question is

The yield for a forest at age $t$ is given by $V=6.7e^{\raisebox{1ex}{$-48.1$}\!\left/ \!\raisebox{-1ex}{$t$}\right.}$

Calculation :

The horizontal asymptote

  $\underset{t\to \infty }{\lim }V\left(t\right)=\frac{6.7}{e^{\frac{48.1}{\infty }}}=\frac{6.7}{e^0}=6.7$

So, the horizontal asymptote of the function is given by substituting the value of x-axis component to $\infty$

c.

To determine

To calculate : the time necessary to obtain a yield of $1.3$ meter cubic field

Answer to Problem 150E

The time necessary to obtain a yield of $1.3$ meter cubic field is

Explanation of Solution

Given information : The question is

The yield for a forest at age $t$ is given by $V=6.7e^{\raisebox{1ex}{$-48.1$}\!\left/ \!\raisebox{-1ex}{$t$}\right.}$

Calculation :

For $1.3$ meter cubic field

  $\begin{array}{l}V\left(t\right)=1.3\\ \frac{-48.1}{t}=\ln \left(\frac{1.3}{6.7}\right)\\ t=29.335\end{array}$