To find the height of the tree using the concept of similar triangles, we can set up a proportion based on the height of the man and the length of his shadow compared to the height of the tree and the length of its shadow.
Let \( h \) represent the height of the tree in inches. The relevant dimensions are as follows:
- Height of the man: 70 inches
- Length of the man's shadow: 80 inches
- Length of the tree's shadow: 164 inches
Since the triangles formed by the man and the tree with their shadows are similar, we can set up the proportion:
\[ \frac{\text{Height of the man}}{\text{Length of the man's shadow}} = \frac{\text{Height of the tree}}{\text{Length of the tree's shadow}} \]
Substituting the known values into this proportion gives us:
\[ \frac{70}{80} = \frac{h}{164} \]
Next, we can cross-multiply to solve for \( h \):
\[ 70 \cdot 164 = 80 \cdot h \]
Calculating \( 70 \cdot 164 \):
\[ 70 \cdot 164 = 11480 \]
Now substituting back into the equation:
\[ 11480 = 80h \]
To isolate \( h \), divide both sides by 80:
\[ h = \frac{11480}{80} \]
Calculating \( \frac{11480}{80} \):
\[ h = 143.5 \]
Thus, the height of the tree is
\[ \boxed{143.5} \text{ inches.} \]
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