To find the height of a tree a surveyor stands 112 ft from t

To find the height of a tree a surveyor stands 112 ft from the base of the tree, and measures the angle of elevation of the line of sight to the top of the tree to be 20.6°. How tall is the tree? Round to the nearest tenth of a foot.

Solution

There is one information missing in the question which is height of surveyor. Because the line of sight is from eyes of surveyor and the height of tree which will come out of the following calculation will be from the eyes of surveyor if we want to factor in the exact numbers.

\\\\tan\\theta=\\frac{height\\ of\\ tree\\from\\ eyes}{horizontal\\ distance\\ from\\ tree}\\\\

\\\\tan(20.6^{\\circ})=\\frac{height\\ of\\ tree\\from\\ eyes}{112}\\\\

\\\\height=tan(20.6^{\\circ})*112=42.01\\ feet\\\\

\\\\height\\ of\\ tree\\ from\\ ground =height\\ from\\ eyes + eyes\\ to\\ ground\\ height\\\\

\\\\height\\ of\\ tree\\ from\\ ground =42.01+5.00=47.01feet\\\\

\\\\assumption\\ of eyes\\ to\\ ground\\ height\\ is\\ taken\\ as\\ 5\\\\