a vertical pole 6 feet high casts a shadow 4 feet long. at the same time a tree casts a shadow 64 feet long. what is the height in feet of the tree?

To find the height of the tree, we can use the concept of similar triangles. The ratio of the height of the pole to the length of its shadow will be the same as the ratio of the height of the tree to the length of its shadow.

Let \( h \) be the height of the tree. According to the problem:

\[
\frac{\text{Height of the pole}}{\text{Length of the pole's shadow}} = \frac{\text{Height of the tree}}{\text{Length of the tree's shadow}}
\]

Substitute the known values:

\[
\frac{6}{4} = \frac{h}{64}
\]

Now solve for \( h \):

\[
\frac{6}{4} = \frac{h}{64}
\]

Multiply both sides by 64 to isolate \( h \):

\[
h = \frac{6}{4} \times 64
\]

Simplifying:

\[
h = 1.5 \times 64 = 96
\]

Thus, the height of the tree is \( \boxed{96} \) feet.

Would you like further details or explanations? 

Here are some related questions:
1. What is the concept of similar triangles?
2. How can you use proportions to solve problems like this?
3. What other real-world applications involve similar triangles?
4. How would this change if the sun's angle changed?
5. How does this method apply to other geometrical shapes?

**Tip**: Always look for proportions when dealing with similar shapes or figures in geometry!

A vertical pole 6 feet high casts a shadow 4 feet long. At the same time, a tree casts a shadow 64 feet long. What is the height in feet of the tree?

Discover how to find the height of a tree by applying the concept of similar triangles. Learn how to set up and solve proportions using a vertical pole casting a shadow as a real-world example. This solution is suitable for students in Grades 6-8.

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