Question

A man who is 70 inches tall has a shadow that is 80 inches long. A nearby tree casts a shadow that is 164 inches long. The shadow of the man and the shadow of the tree form similar triangles. How tall is the tree in inches?(1 point)

Answer 1

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.