This problem involves similar triangles, as the flagpole and its shadow, and the tree and its shadow, form right triangles with the same angles. To find the length of the shadow cast by the tree, we can set up a proportion based on the heights and shadow lengths of the two objects.

Let the length of the shadow cast by the tree be $x$ meters.

We can write the proportion as: $\frac{\text{Height of Flagpole}}{\text{Length of Flagpole's Shadow}} = \frac{\text{Height of Tree}}{\text{Length of Tree's Shadow}}$ Substituting the given values: $\frac{15}{50} = \frac{27}{x}$

Now, solve for $x$: $15x = 27 \times 50$ $15x = 1350$ $x = \frac{1350}{15} = 90$

So, the length of the shadow cast by the tree is $90$ meters.

The shadow cast by a 27 meters high tree is 90 meters.

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How would the shadow change if the tree were 35 meters tall?
2. What would the shadow length be if the flagpole's height increased to 20 meters?
3. If the sun's angle changes, how might it affect the shadow lengths?
4. Can you use this method to find the height of an object based on its shadow?
5. What is the role of the angle of elevation of the sun in determining shadow length?

Tip: Whenever two objects cast shadows under the same lighting conditions, you can use similar triangles to solve for unknown lengths.