Chapter 1.3, Problem 38E

To determine

To sketch: The graph of the function by plotting the height h on the vertical axis and the age in decades on the horizontal axis.

Answer to Problem 38E

The graph of the function by plotting the height h on the vertical axis and the age in decades on the horizontal axis as given below,

  

Explanation of Solution

Given information:

The height of the tree $h$ is given by,

  $h=\frac{100a}{100+a}$ ,where $a$ represents the age of the tree in years

Calculation:

The height of the tree $h$ (in meters) obeys

  $h=\frac{100a}{100+a}\mathrm{................}(A)$

Where a represents age of the tree in years for

  $0\le a\le 1000$

Now, for $a=200,a=400,a=600,a=800,$ we have the height of the tree $\left(h\right)$ as follows:

For

  $\begin{array}{l}a=200\\ h=\frac{100\left(200\right)}{100+200}\\ =\frac{20000}{300}\\ h=66.67\end{array}$

For

  $\begin{array}{l}a=400\\ h=\frac{100\left(400\right)}{100+400}\\ =\frac{40000}{500}\\ h=80\\ \end{array}$

For $a=600$

  $\begin{array}{l}a=600\\ h=\frac{100\left(600\right)}{100+600}\\ =\frac{60000}{700}\\ h=85.72\\ \end{array}$

For $\begin{array}{l}a=800\\ \end{array}$

  $\begin{array}{l}a=800\\ h=\frac{100\left(800\right)}{100+800}\\ =\frac{80000}{900}\\ h=88.88\\ \end{array}$

For $a=1000$

  $\begin{array}{l}a=1000\\ h=\frac{100\left(1000\right)}{100+1000}\\ =\frac{100000}{1100}\\ h=90.90\\ \end{array}$

We can draw the graph of the function $\left(A\right)$ by plotting the height $h$ on the vertical axis and age of trees on the horizontal axis as given below as,

  

Let

  $d$ be the age of tree in decades.

Then we have

  $\begin{array}{l}d=10a\\ a=\frac{d}{\begin{array}{l}10\\ \end{array}}\end{array}$

Hence, height of the tree $\left(h\right)$ in terms of decades $\left(d\right)$ is as follows

  $\begin{array}{l}h=\frac{100\left(\frac{d}{10}\right)}{100+\left(\frac{d}{10}\right)}\\ =\frac{100d}{1000+d}\end{array}$

For $d=200$

  $\begin{array}{l}h=\frac{100(200)}{1000+200}\\ =\frac{20000}{1200}\\ =16.67\end{array}$

For $d=400$

  $\begin{array}{l}h=\frac{100(400)}{1000+400}\\ =\frac{40000}{1400}\\ =28.57\end{array}$

For $d=600$

  $\begin{array}{l}h=\frac{100(600)}{1000+600}\\ =\frac{60000}{1600}\\ =37.5\end{array}$

For $d=800$

  $\begin{array}{l}h=\frac{100(800)}{1000+800}\\ =\frac{80000}{1800}\\ =44.44\end{array}$

For $d=1000$

  $\begin{array}{l}h=\frac{100(1000)}{1000+1000}\\ =\frac{100000}{2000}\\ =50\end{array}$

The graph of the function by plotting the height h on the vertical axis and the age in decades on the horizontal axis as given below,