To find the distance between the tourist on the observation deck and her friend on the ground, we need to use trigonometry.
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Understand the Problem:
- The height of the observation deck is \( h = 520 \) ft.
- The angle of depression from the deck to her friend is \( \theta = 80.2° \).
- We need to find the horizontal distance (\( d \)) between the tourist directly above her friend and the friend on the ground.
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Apply the Trigonometric Function: The angle of depression from the horizontal line means that we can visualize this as an angle formed with respect to the height and distance on the ground below. Since this forms a right triangle, we can use the tangent function.
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} \]
Rearranging gives us:
\[ d = \frac{h}{\tan(\theta)} \]
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Calculate \( d \): Plugging in the values, we get:
\[ d = \frac{520 \text{ ft}}{\tan(80.2°)} \]
We'll use a calculator to find \( \tan(80.2°) \).
\[ \tan(80.2°) \approx 5.144 \]
So now, we can compute \( d \):
\[ d = \frac{520 \text{ ft}}{5.144} \approx 101.1 \text{ ft} \]
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Use the Pythagorean Theorem to Find the Distance: Now, we need to find the actual straight-line distance (\( s \)) between the tourist and her friend using the Pythagorean theorem:
\[ s = \sqrt{h^2 + d^2} \]
Calculating \( d \):
\[ d = \frac{520}{5.144} \approx 101.1 \text{ ft} \]
Now substituting \( h \) and \( d \):
\[ s = \sqrt{(520)^2 + (101.1)^2} \]
\[ s = \sqrt{270400 + 10222.42} \approx \sqrt{280622.42} \approx 529.2 \text{ ft} \]
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Rounding Off: Rounding \( 529.2 \) to the nearest tenth gives us \( 529.2 \) ft. However, we should check the options.
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Final Answer: Among the options provided, the closest one is:
527.7 ft.
Thus, the final answer is:
527.7 ft.
1 answer