To find the height of the tree, we can use the properties of similar triangles. Since the man and the tree cast shadows that are proportional, we can set up a proportion based on their heights and shadow lengths.
Let \( h \) be the height of the tree. The proportions from the similar triangles can be set up as follows:
\[ \frac{\text{Height of the man}}{\text{Length of the man's shadow}} = \frac{\text{Height of the tree}}{\text{Length of the tree's shadow}} \]
Now we can substitute in the known values:
\[ \frac{70}{80} = \frac{h}{164} \]
Now we will cross-multiply to solve for \( h \):
\[ 70 \cdot 164 = 80 \cdot h \]
Calculating \( 70 \cdot 164 \):
\[ 70 \cdot 164 = 11480 \]
Now the equation becomes:
\[ 11480 = 80h \]
Now, divide both sides by 80 to solve for \( h \):
\[ h = \frac{11480}{80} = 143.5 \]
Therefore, the height of the tree is \( \boxed{143.5} \) inches.
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