We can use the properties of similar triangles to solve this problem.
Let’s denote:
- the height of the man as \( H_m = 70 \) inches,
- the length of the man’s shadow as \( S_m = 80 \) inches,
- the height of the tree as \( H_t \) (which we want to find),
- and the length of the tree’s shadow as \( S_t = 164 \) inches.
Since the triangles are similar, we can set up a proportion based on the heights and the lengths of the shadows:
\[ \frac{H_m}{S_m} = \frac{H_t}{S_t} \]
Substituting in the known values:
\[ \frac{70}{80} = \frac{H_t}{164} \]
Now, we can cross-multiply to solve for \( H_t \):
\[ 70 \cdot 164 = 80 \cdot H_t \]
Calculating the left side:
\[ 11480 = 80 \cdot H_t \]
Now, divide both sides by 80 to isolate \( H_t \):
\[ H_t = \frac{11480}{80} = 143.5 \]
Thus, the height of the tree is \( \boxed{143.5} \) inches.