Chapter 7.1, Problem 88E

To determine

To sketch the graph of the two log rules and hence find the diameter by which both the rules agrees.

Answer to Problem 88E

The diameter where both the rules agrees is $D=2.98$

For selling large logs, Scribner rule is used.

Explanation of Solution

Given information: The given information is:

Let $V\text{and}D$ be the volume and the diameter of the log.

The given log rules are:

  $\text{Doyle}\text{log}\text{Rule:}{(D-4)}^2,5\le D\le 40$

  $\text{Scribner}\text{Log Rule:}0.79D^2-2D-4,5\le D\le 40$

Calculation:

(a) Using graphing utility to sketch the system of equations:

  ${(D-4)}^2;0.79D^2-2D-4$

Enter the expressions ${(D-4)}^2$ and $0.79D^2-2D-4$ in the graphing utility.

The graph is given as:

  

From the above graph, it is concluded that both the rules satisfies at $D=2.98$

(b) From the above graph, it is clearly seen that both the log rules intersect at $D=2.98$ , which is the point of mutual consent.

(c)  For selling large logs, the rule which gives the small volume is often in use.

From the above graph, it is clearly seen that Scribner rule has less volume as compared to the Doyle rule.

Hence, for selling large logs, Scribner rule has been used.