Dalvin is 1.85 meters tall. At 3 p.m., he measures the length of a tree's shadow to be 31.35 meters. He stands 27.2 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.

This problem involves using similar triangles to find the height of the tree. Dalvin and the tree create two right triangles with their shadows as one side and their heights as the other.

Let's define the variables:
- Dalvin's height: \( h_D = 1.85 \, \text{meters} \)
- Dalvin's shadow length: \( s_D \, \text{meters} \)
- Tree's height: \( h_T \, \text{meters} \)
- Tree's shadow length: \( s_T = 31.35 \, \text{meters} \)

From the problem, we know the total shadow length is 31.35 meters. Dalvin is standing 27.2 meters from the tree, so the length of Dalvin's shadow is:
\[
s_D = 31.35 - 27.2 = 4.15 \, \text{meters}
\]

Now, we can use the property of similar triangles: the ratios of corresponding sides are equal. This gives us the proportion:
\[
\frac{h_D}{s_D} = \frac{h_T}{s_T}
\]
Substitute the known values:
\[
\frac{1.85}{4.15} = \frac{h_T}{31.35}
\]
Now, solve for \( h_T \) (the height of the tree):
\[
h_T = \frac{1.85 \times 31.35}{4.15}
\]

Let me calculate that.The height of the tree is approximately \( 13.98 \, \text{meters} \) to the nearest hundredth.

Would you like more details on how this was calculated or have any other questions?

Here are 5 related questions to deepen your understanding:
1. How would the height of the tree change if Dalvin was closer to the tree?
2. What if the length of the tree's shadow was shorter, how would that affect the tree's height?
3. How does the concept of similar triangles help in real-world applications like this one?
4. Can you use this method to find the height of an object if the shadows are not aligned?
5. What other methods could be used to find the height of a tall object using shadows?

**Tip:** Whenever dealing with proportions in geometry, make sure to carefully match corresponding sides or angles to avoid calculation errors!

Learn how to find the height of a tree using similar triangles and shadow proportions. In this example, Dalvin stands 27.2 meters away from the tree and observes the shadow lengths at 3 p.m. A step-by-step guide is provided to solve the problem using basic geometry and proportionality, ideal for students in Grades 8-10.

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