ENTS,

    Problem #7 is attached. I will draw diagrams for scenarios 2 and 3 if anyone wants them. We are grateful to Beth for suggestion the inclusion of this problem into the problem set.

Bob

Problem #7: Beth Koebel sets out to measure an oak in a ravine. However she has trouble finding a spot where she can see both the top and bottom of the tree. She notes a branch stub from a position where she sees the top. She sees the same stub from a spot where the base is visible. How may Beth use the stub to measure the full height of the tree?

Solution: The diagram below shows the solution where the branch stub is above eye level from both vantage points. The method of solution is to divide the tree into two sections: top down to the stub (identified as point P in the diagram) and from P down to the base of the oak. The equation in the diagram consists of four terms. The first two terms are: (1) the height from the top down to eye level, and (2) the height from point P down to eye level. These measurements are made from vantage point E₁. The second two terms are: (1) the height of P above eye level, and (2) the height from the base up to eye level. These measurements are made from vantage point E₂. The four quantities are algebraically combined as shown in the formula, where H stands for total height of the tree.

            In this kind of problem, there are actually three scenarios to consider. They are listed below.

Scenario #1: Point P is above eye level from both vantage points. The equation is

 The above diagram depicts this first scenario and it is probably the most common one.

Scenario #2: Point P is below eye level from both vantage points. The equation is

Scenario #3: Point P is below eye level from the higher vantage points and above eye level from the lower vantage point. The equation is

Comments: This problem illustrates a common situation that is often encountered in the field within the interior of a forest, i.e. no common spot where the top and bottom can be seen. In these cases, the tree must be divided up into portions that can be individually measured. The separate measurements are then combined. Also, situations do occur where positions E₁ and E₂ occur on the same horizontal plane. The measurer shifts forward or backward to view first top and then the bottom. The method of solution is the normal, i.e. sine top – sine bottom. Diagrams similar to the one above can be drawn to clarify each of the above scenarios.