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?

Answer 1

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

\[
\boxed{143.5} \text{ inches.}
\]