Math Problem Statement

Adrian is 1.65 meters tall. At 11 a.m., he measures the length of a tree's shadow to be 31.25 meters. He stands 27 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.

Solution

This problem involves similar triangles, where the proportions between the heights and shadow lengths of Adrian and the tree are equal. Here's how to solve it step by step:


Step 1: Understand the situation

  • Adrian's height: hA=1.65mh_A = 1.65 \, \text{m}
  • Adrian's shadow length: xmx \, \text{m}
  • Tree's shadow length: 31.25m31.25 \, \text{m}
  • Distance between Adrian and the tree: 27m27 \, \text{m}
  • Total shadow length (tree + Adrian's shadow): 31.25m31.25 \, \text{m}

Step 2: Relate Adrian's shadow length to the total shadow length

Since Adrian and the tree's shadows meet at the same point, Adrian's shadow length is the total shadow length minus the distance between him and the tree: x=31.2527=4.25m.x = 31.25 - 27 = 4.25 \, \text{m}.


Step 3: Use the similarity of triangles

The triangles formed by Adrian and his shadow and by the tree and its shadow are similar. Therefore, their corresponding ratios are equal: Adrian’s heightAdrian’s shadow length=Tree’s heightTree’s shadow length.\frac{\text{Adrian's height}}{\text{Adrian's shadow length}} = \frac{\text{Tree's height}}{\text{Tree's shadow length}}. Substituting the known values: 1.654.25=hT31.25.\frac{1.65}{4.25} = \frac{h_T}{31.25}.


Step 4: Solve for the tree's height (hTh_T)

Rearranging for hTh_T: hT=1.654.2531.25.h_T = \frac{1.65}{4.25} \cdot 31.25. First, calculate 1.654.25\frac{1.65}{4.25}: 1.654.25=0.3882(approximately).\frac{1.65}{4.25} = 0.3882 \, (\text{approximately}). Now, multiply this by 31.2531.25: hT=0.388231.25=12.14m.h_T = 0.3882 \cdot 31.25 = 12.14 \, \text{m}.


Final Answer:

The height of the tree is approximately 12.14 meters.


Do you want me to explain this further or help with a different example? Here are some related questions to explore:

  1. How can the concept of similar triangles apply in real-life scenarios other than shadows?
  2. What happens if Adrian stands closer or farther from the tree? How would the calculations change?
  3. How would you solve this problem if the shadow lengths were unknown, but the height of the tree was given instead?
  4. Can you derive the formula for finding the height of a tree using any person's height and shadow length?
  5. What is the role of proportionality in scaling or measurements in geometry?

Tip: Always draw a diagram to visualize problems involving geometry and similar triangles—it makes the proportions clearer.

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Math Problem Analysis

Mathematical Concepts

Geometry
Proportionality
Similar Triangles

Formulas

Shadow length relationship: Total shadow length = Tree's shadow length + Adrian's shadow length
Proportionality formula: (Height of object) / (Shadow length of object) = (Height of another object) / (Shadow length of another object)

Theorems

Similarity of Triangles

Suitable Grade Level

Grades 8-10