To solve the problem, we need to set up the situation using basic trigonometry. Let's define some variables and work step by step.
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Define the variables:
- Let \( h \) = height of the flag pole.
- Let \( d \) = distance from point A to the base of the flag pole.
- We know the angle of elevation from point A is \( 13^\circ \) and from point B it is \( 30^\circ \).
- The distance between point A and point B is \( 120 \) meters.
Therefore, we can write: \[ d - 120 = \text{distance from point A to point B} \]
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Using the angle from point A: From point A: \[ \tan(13^\circ) = \frac{h}{d} \] Rearranging gives us: \[ h = d \cdot \tan(13^\circ) \]
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Using the angle from point B: From point B, whose distance from the flag pole is \( d - 120 \): \[ \tan(30^\circ) = \frac{h}{d - 120} \] Rearranging gives us: \[ h = (d - 120) \cdot \tan(30^\circ) \]
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Setting the equations equal: We have two expressions for \( h \): \[ d \cdot \tan(13^\circ) = (d - 120) \cdot \tan(30^\circ) \]
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Solving for \( d \): Expanding and rearranging: \[ d \cdot \tan(13^\circ) = d \cdot \tan(30^\circ) - 120 \cdot \tan(30^\circ \] \[ d \cdot \tan(13^\circ) - d \cdot \tan(30^\circ) = -120 \cdot \tan(30^\circ) \] Factoring out \( d \): \[ d (\tan(13^\circ) - \tan(30^\circ)) = -120 \cdot \tan(30^\circ) \] Therefore: \[ d = \frac{-120 \cdot \tan(30^\circ)}{\tan(13^\circ) - \tan(30^\circ)} \]
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Calculating \( d \): Using \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.5774 \) and \( \tan(13^\circ) \approx 0.2309 \): \[ d = \frac{-120 \cdot \frac{1}{\sqrt{3}}}{0.2309 - 0.5774} \] \[ d = \frac{-120 \cdot \frac{1}{\sqrt{3}}}{-0.3465} \approx \frac{120 \cdot 0.5774}{0.3465} \approx \frac{69.288}{0.3465} \approx 200.0 \]
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Finding \( h \) using \( d \): Now substitute \( d \) back into \( h = d \cdot \tan(13^\circ) \): \[ h \approx 200.0 \cdot 0.2309 \approx 46.18 \]
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Finding the distance from point B to the top of the flag post: From point B, the height remains 46.18 meters, and the base distance from B to the pole is: \[ d - 120 = 200 - 120 = 80 \text{ meters} \] Now we can find the distance from point B to the top of the flag post using the Pythagorean theorem: \[ \text{Distance from B to top} = \sqrt{h^2 + (d - 120)^2} = \sqrt{(46.18)^2 + (80)^2} \] \[ = \sqrt{2135.056 + 6400} \approx \sqrt{8535.056} \approx 92.58 \text{ meters} \]
Thus, the distance from point B to the top of the flag post is approximately 92.58 meters.
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