Fatoumata is 1.55 meters tall. At 11 a.m., she measures the length of a tree's shadow to be 21.25 meters. She stands 17 meters away from the tree, so that the tip of her shadow meets the tip of the tree's shadow. Find the height of the tree to the nearest hundredth of a meter.

This problem involves similar triangles, where Fatoumata and the tree form one triangle, and their shadows form another triangle. Since both triangles share the same angles, they are similar, which allows us to set up a proportion.

Here’s how we solve it step by step:

### Given:
- Fatoumata's height \( h_F = 1.55 \) meters.
- Fatoumata's distance from the tree (where their shadows meet) is 17 meters.
- The tree's shadow is 21.25 meters.
  
Let's define:
- \( h_T \) as the height of the tree (the unknown).
- Fatoumata’s shadow length can be found as the total distance of 21.25 meters minus her distance from the tree:  
  \[
  \text{Fatoumata's shadow} = 21.25 - 17 = 4.25 \, \text{meters}.
  \]

Now, using the property of similar triangles, we can set up the following proportion:

\[
\frac{h_F}{\text{Fatoumata's shadow}} = \frac{h_T}{\text{Tree's shadow}}.
\]

Substituting the values:

\[
\frac{1.55}{4.25} = \frac{h_T}{21.25}.
\]

To solve for \( h_T \), we cross-multiply:

\[
h_T = \frac{1.55 \times 21.25}{4.25}.
\]

Now, calculating the height of the tree:

\[
h_T = \frac{32.9375}{4.25} = 7.75 \, \text{meters}.
\]

### Answer:
The height of the tree is approximately **7.75 meters**.

Would you like more details or have any questions about the process?

Here are five related questions to deepen your understanding:
1. How do we know the triangles are similar in this problem?
2. What would change if Fatoumata were standing at a different distance from the tree?
3. Can you calculate the length of Fatoumata's shadow in a different scenario using similar triangles?
4. How does this problem relate to the concept of proportionality in geometry?
5. How could you apply this technique to other real-world situations?

**Tip:** Always check if a problem involves similar triangles when objects cast shadows at the same time and angle!

Learn how to find the height of a tree using Fatoumata's height and shadow measurements. This problem explores similar triangles and proportionality concepts in geometry. The step-by-step solution includes cross-multiplication to solve for the height of the tree, suitable for students in Grades 6-8.

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