https://file.aiquickdraw.com/mathbot/file/17276855303787p7gpeqv.jpg

To solve this problem, we'll use the fact that the student and the tree, along with their respective shadows, form similar triangles. The key here is to set up a proportion using the corresponding parts of the triangles: the height of the student to the length of her shadow, and the height of the tree to the length of the tree's shadow.

1. **Student’s height**: The student is 5 feet 6 inches, which is 5.5 feet.
2. **Student’s shadow**: The shadow is 6 feet long.
3. **Tree’s shadow**: The shadow is 20 feet long.
4. We are asked to find the height of the tree (let’s call it **h**).

Using the property of similar triangles, we can set up the proportion:

\[
\frac{\text{Student's height}}{\text{Student's shadow}} = \frac{\text{Tree's height}}{\text{Tree's shadow}}
\]

Substitute the known values:

\[
\frac{5.5}{6} = \frac{h}{20}
\]

Now, solve for **h**:

\[
h = \frac{5.5 \times 20}{6} = \frac{110}{6} = 18.33 \, \text{ft}
\]

Thus, the height of the tree is **18.33 feet**.

Let me know if you have any questions or need further details!

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### Related Questions:
1. How does the concept of similar triangles apply to other real-world situations?
2. What would happen if the student's shadow was longer than 6 feet?
3. How can we apply this method to find the height of other objects, like buildings or poles?
4. What other methods could be used to find the height of a tree?
5. If the tree's shadow was shorter, would the tree still be taller than the student?

### Tip:
To solve similar triangle problems, always make sure to set up the correct proportion, ensuring that corresponding sides are compared (height to height, shadow to shadow).

A 5'6" student casts a shadow 6 ft long at the same time a tree casts a 20 ft shadow. How tall is the tree? (Hint: The tree and its shadow are similar to the student and her shadow.)

Learn how to calculate the height of a tree using similar triangles and proportions. In this problem, a 5'6" student casts a shadow 6 ft long while a tree casts a 20 ft shadow. Using geometry concepts, this problem is suitable for students in Grades 8-10.

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