Bot GPT 4-o mini To find the angle of elevation \(\theta\) from a point on the ground to the top of a flagpole, we can use the tangent function from trigonometry. The tangent of the angle is the ratio of the opposite side (the height of the flagpole) to the adjacent side (the distance from the point to the flagpole).
In this case, the height of the flagpole (opposite side) is 31.9 m, and the distance from the point on the ground to the base of the flagpole (adjacent side) is 55 m. The relationship can be expressed as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{31.9}{55} \]
To find the angle \(\theta\), we will take the inverse tangent (arctan) of this ratio:
\[ \theta = \arctan\left(\frac{31.9}{55}\right) \]
Calculating this ratio:
\[ \frac{31.9}{55} \approx 0.58 \]
Now, using a calculator to find the arctangent:
\[ \theta \approx \arctan(0.58) \approx 29.7^\circ \]
Thus, the angle of elevation of the top of the flagpole is approximately \(29.7^\circ\).
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