Question

Jeriel is 1.65 meters tall. At 2 p.m., he measures the length of a tree's shadow to be 28.05 meters. He stands 23.5 meters away from the tree, so that the tip of his shadow meets the tip of the tree's shadow. Find the height of the tree to the nearest hundredth of a meter.
23.5 m
28.05 m
 1.65 m
(Diagram is not to scale.)
Answer
Attempt 1 out of 2

Answer 1

To find the height of the tree, we can use the principle of similar triangles. The two triangles we can consider here are the one formed by Jeriel and his shadow, and the one formed by the tree and its shadow.

Let \( h \) be the height of the tree.

### Given:
- Height of Jeriel: \( H_j = 1.65 \) m
- Length of Jeriel's shadow: \( S_j = 28.05 \) m
- Distance from Jeriel to the tree: \( D = 23.5 \) m

### Total length of tree's shadow:
The total length of the tree's shadow is the distance from Jeriel to the tree added to the length of Jeriel's shadow:
\[
S_t = S_j + D = 28.05 \, \text{m} + 23.5 \, \text{m} = 51.55 \, \text{m}
\]

### Using similar triangles:
The ratios of the heights to the lengths of the shadows can be set up as follows:
\[
\frac{H_j}{S_j} = \frac{h}{S_t}
\]
Substituting the values we know:
\[
\frac{1.65}{28.05} = \frac{h}{51.55}
\]

### Cross-multiplying to solve for \( h \):
\[
1.65 \cdot 51.55 = h \cdot 28.05
\]

\[
h = \frac{1.65 \cdot 51.55}{28.05}
\]

Calculating the right side:
\[
h = \frac{85.6575}{28.05} \approx 3.055
\]

### Rounding to the nearest hundredth:
\[
h \approx 3.06 \, \text{m}
\]

So, the height of the tree is approximately **3.06 meters**.