"Published in Newark, California, USA"
A woodman makes a wedge-shaped cut in the trunk of a tree. Assume that the trunk is a right circular cylinder of radius 4 in., that the lower surface of the cut is a horizontal plane, and that the upper surface is a plane inclined at an angle of 45° to the horizontal and intersecting the lower surface of the cut in a diameter. This wedge is then cut into two equal pieces by a vertical cut. What is the area of this vertical section? The woodman now wishes to divide one of these pieces by another section parallel to the first section. If the new section is to have an area equal to one-fourth that of the original section, where should the cut be made?
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
If you cut a wedge at its center, then the section is an isosceles right triangle. Therefore, the area of a section is
If you cut the other part of the wedge parallel to the first section, then the other section is also an isosceles right triangles. Hence, the two isosceles right triangles are similar. Let's consider the cross section of the trunk as follows
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| Photo by Math Principles in Everyday Life |
By using similar triangles, the length of the base of the other isosceles right triangle is
Therefore, by Pythagorean Theorem, the distance of the other section from the first section is
