To find the height of the flagpole \( x \) and the distance \( y \) from you to the top of the flagpole, we can use trigonometry.
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Find the height of the flagpole \( x \): We can use the tangent function, which relates the angle of elevation to the opposite side (the height of the pole) and the adjacent side (the distance from you to the base of the pole).
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Given:
- \( \theta = 40^\circ \)
- adjacency = 25 meters
Thus, we can write:
\[ \tan(40^\circ) = \frac{x}{25} \]
Rearranging gives:
\[ x = 25 \tan(40^\circ) \]
Calculating \( \tan(40^\circ) \):
\[ \tan(40^\circ) \approx 0.8391 \]
Therefore:
\[ x \approx 25 \times 0.8391 \approx 20.9775 \]
Rounding to the nearest meter, the height of the flagpole \( x \approx 21 \) meters.
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Find the distance \( y \) from you to the top of the flagpole: The distance to the top of the flagpole can be calculated using the sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Here, we use:
\[ \sin(40^\circ) = \frac{x}{y} \]
Rearranging gives:
\[ y = \frac{x}{\sin(40^\circ)} \]
We already found \( x \approx 20.9775 \).
Calculating \( \sin(40^\circ) \):
\[ \sin(40^\circ) \approx 0.6428 \]
Thus:
\[ y \approx \frac{20.9775}{0.6428} \approx 32.6776 \]
Rounding to the hundredths place gives \( y \approx 32.68 \) meters.
Final Answers:
- Height of the flagpole \( x \approx 21 \) meters.
- Distance to the top of the flagpole \( y \approx 32.68 \) meters.
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