Question Joseph is 1.75 meters tall. At $$ 11 \mathrm{a} . \mathrm{m} $$. ., he measures the length of a tree's shadow to be 34.05 meters. He stands 29.7 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.

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To determine the height of the tree, we'll use the principle of similar triangles. Here's a step-by-step breakdown:

1. **Understanding the Scenario:**
   - **Joseph's Height:** 1.75 meters.
   - **Tree's Shadow Length:** 34.05 meters.
   - **Distance from Tree to Joseph:** 29.7 meters.
   - **Condition:** The tip of Joseph's shadow meets the tip of the tree's shadow.

2. **Visualizing the Setup:**
   - The tree casts a shadow of 34.05 meters.
   - Joseph stands 29.7 meters away from the tree along the direction of the shadow.
   - This means Joseph's shadow extends $$ 34.05 \, \text{m} - 29.7 \, \text{m} = 4.35 \, \text{m} $$.

3. **Applying Similar Triangles:**
   - The ratio of the height of the tree to its shadow is the same as the ratio of Joseph's height to his shadow.
   $$
   \frac{\text{Height of Tree}}{\text{Shadow of Tree}} = \frac{\text{Height of Joseph}}{\text{Shadow of Joseph}}
   $$
   $$
   \frac{H}{34.05} = \frac{1.75}{4.35}
   $$
   
4. **Solving for the Height of the Tree ($$ H $$):**
   $$
   H = \frac{34.05 \times 1.75}{4.35}
   $$
   $$
   H \approx \frac{59.5875}{4.35} \approx 13.70 \, \text{meters}
   $$

**Final Answer:**  
The height of the tree is **13.70 meters**.
The tree is approximately 13.70 meters tall.

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