Similar triangles are triangles that have the same shape but different sizes. This means their corresponding angles are equal, and their corresponding sides are proportional. This concept is crucial for solving problems involving heights and shadows....
Applying Similar Triangles to the Problem
In this scenario, we have two triangles: one formed by Joseph and his shadow, and the other formed by the tree and its shadow. These triangles are similar because the sun's rays hit Joseph and the tree at the same angle, creating parallel lines.
Let's visualize the scenario:
/|\
/ | \
/ | \
/ | \
/ | \
/_____|_____\
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29.7m | 34.05m
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Joseph Tree
We know Joseph's height (1.75 meters) and the length of his shadow. We also know the distance between Joseph and the tree, and the length of the tree's shadow. This information allows us to set up a proportion to find the height of the tree.
Setting Up the Proportion
The ratio of Joseph's height to his shadow's length will be equal to the ratio of the tree's height to its shadow's length. This is represented by the following equation:
Joseph's Height / Joseph's Shadow Length = Tree's Height / Tree's Shadow Length
Plugging in the known values:
1.75 meters / 29.7 meters = Tree's Height / 34.05 meters
Solving for the Tree's Height
To solve for the tree's height, we can cross-multiply and then isolate the unknown variable:
1.75 meters * 34.05 meters = 29.7 meters * Tree's Height
59.59 meters2 = 29.7 meters * Tree's Height
Tree's Height = 59.59 meters2 / 29.7 meters
Tree's Height ≈ 2.00 meters
Final Answer
Therefore, the height of the tree is approximately **2.00 meters** to the nearest hundredth of a meter.