To find the height of the tree, we can use the information given and apply trigonometric ratios.
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Convert Shaina's height: Shaina is 5 feet 6 inches tall. We convert this entirely to inches: \[ 5 \text{ feet} = 5 \times 12 = 60 \text{ inches} \] \[ 5 \text{ feet} 6 \text{ inches} = 60 + 6 = 66 \text{ inches} \]
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Identify known values:
- Distance from Shaina to the base of the tree = 20 feet
- Angle of elevation to the bird = 68 degrees
- Height of Shaina = 66 inches (or 5.5 feet)
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Determine the total height of the tree: Let \( h \) be the height of the tree in feet above Shaina's height.
We can use the tangent function, which relates the angle of elevation to the opposite and adjacent sides: \[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \] In this case, the opposite side is the height of the tree above Shaina (\( h \)), and the adjacent side is the distance from Shaina to the tree (20 feet): \[ \tan(68^\circ) = \frac{h}{20} \] Rearranging gives: \[ h = 20 \cdot \tan(68^\circ) \]
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Calculate \( h \): Using a calculator, find \( \tan(68^\circ) \): \[ \tan(68^\circ) \approx 2.4751 \] Now we substitute this back: \[ h = 20 \cdot 2.4751 \approx 49.502 \text{ feet} \]
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Total height of the tree: To find the total height of the tree, we need to add Shaina's height (5.5 feet) to the height \( h \): \[ \text{Total height of the tree} = h + \text{height of Shaina} = 49.502 + 5.5 \approx 55.002 \text{ feet} \]
Since the question asks for the height of the tree in feet, rounding it gives approximately 55 feet.
So the height of the tree is: 55 feet.
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