In its September/October 1998 issue, Workbench, had an article on "Felling a Tree Safely." Part of the process of felling a tree is to cut a notch on the side of the tree toward the direction you want it to fall. This notch is opposite the cut line to be made in felling the tree and its crease is an inch or two below the cut line.
Workbench provided pictures and a caption that said " From the front, the notch is about 80% of the tree's diameter. From the side, its depth is about 1/3 of the tree's diameter."
Problem: Can the crease length be 80 % of the tree's diameter and the depth of the cut be one-third of the tree's diameter?
Consider a cross-section of the tree trunk through the crease line. The crease line will be a chord of the circle (cross section) and the depth will be measured along a diameter perpendicular to the crease.
Let CD represent the crease, FE the diameter, and GE the depth of the cut. Draw CF and CE.
FCE is a right triangle. Therefore,
Now, CG is half of the chord CD and FE is the diameter. This means that the length of the crease would be a little more that 94 % of the diameter -- not very close to "about 80 %."
What fraction of the diameter should be the depth of the cut if the crease length is 80 % of the diameter?
Hint -- if needed.