To determine the height of the waterfall, we can use the concept of similar triangles.
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Let's outline the scenario:
- The hiker is 6 feet tall.
- The distance from the hiker to the mirror is 7.5 feet.
- The distance from the base of the waterfall to the mirror is 35 feet.
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The setup creates two similar triangles:
- Triangle 1 (the hiker's triangle): The height of the hiker is 6 feet, and the distance from the hiker to the mirror is 7.5 feet.
- Triangle 2 (the waterfall's triangle): The height we want to find, which we'll call \( h \), and the distance from the waterfall to the mirror is 35 feet.
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Set up the proportion based on the similarity of the triangles:
\[ \frac{\text{Height of Hiker}}{\text{Distance from Hiker to Mirror}} = \frac{\text{Height of Waterfall}}{\text{Distance from Waterfall to Mirror}} \]
Plugging in the known values:
\[ \frac{6 \text{ ft}}{7.5 \text{ ft}} = \frac{h}{35 \text{ ft}} \]
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Cross-multiply to solve for \( h \):
\[ 6 \cdot 35 = 7.5 \cdot h \]
\[ 210 = 7.5h \]
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Divide both sides by 7.5 to isolate \( h \):
\[ h = \frac{210}{7.5} \]
Simplifying:
\[ h = 28 \text{ ft} \]
Therefore, the height of the waterfall is 28 feet.