To find the height of the flagpole \( x \) and the distance from you to the top of the flagpole \( y \), we can use some basic trigonometry.
- Find the height of the flagpole \( x \):
Given:
- Distance from you to the flagpole (horizontal distance, adjacent side) = 25 meters
- Angle of elevation = 40 degrees
Using the tangent function, which is defined as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
we can express the height of the flagpole \( x \) as:
\[ \tan(40^\circ) = \frac{x}{25} \]
Rearranging this equation to solve for \( x \):
\[ x = 25 \cdot \tan(40^\circ) \]
Now, calculate \( x \):
Using a calculator, we find:
\(\tan(40^\circ) \approx 0.8391\)
So:
\[ x \approx 25 \cdot 0.8391 \approx 20.9775 \]
Rounding to the nearest meter:
\[ x \approx 21 \text{ meters} \]
- Find the distance \( y \) from you to the top of the flagpole:
To find the distance \( y \), we can use the cosine function:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Here, the hypotenuse \( y \) relates to the height of the flagpole and the distance to it:
\[ \cos(40^\circ) = \frac{25}{y} \]
Rearranging to find \( y \):
\[ y = \frac{25}{\cos(40^\circ)} \]
Calculating \( \cos(40^\circ) \):
\(\cos(40^\circ) \approx 0.7660\)
Substituting this value:
\[ y \approx \frac{25}{0.7660} \approx 32.65 \]
Rounding to the hundredths place:
\[ y \approx 32.65 \text{ meters} \]
Final Answers:
- \( x = 21 \) meters
- \( y \approx 32.65 \) meters
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