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Answer :
We start by considering a right triangle where the tree forms the opposite side and the horizontal distance from the tree forms the adjacent side. The angle of elevation, denoted by [tex]$\theta$[/tex], can be determined using the tangent function:
[tex]$$
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
$$[/tex]
Since the tree is [tex]$186$[/tex] feet tall (opposite side) and the distance is [tex]$57$[/tex] feet (adjacent side), we compute:
[tex]$$
\tan \theta = \frac{186}{57} \approx 3.2632
$$[/tex]
To find the angle [tex]$\theta$[/tex], we take the inverse tangent (arctan) of this ratio:
[tex]$$
\theta = \arctan(3.2632)
$$[/tex]
Evaluating [tex]$\arctan(3.2632)$[/tex], we obtain:
[tex]$$
\theta \approx 1.2734 \text{ radians}
$$[/tex]
Since the answer needs to be in degrees, we convert from radians to degrees using the conversion factor [tex]$180^\circ/\pi$[/tex]:
[tex]$$
\theta \approx 1.2734 \times \frac{180^\circ}{\pi} \approx 72.9622^\circ
$$[/tex]
Finally, rounding the result to the nearest tenth of a degree gives:
[tex]$$
\theta \approx 73.0^\circ
$$[/tex]
Thus, the angle of elevation from level ground measured from [tex]$57$[/tex] feet away is approximately [tex]$\boxed{73.0^\circ}$[/tex].
[tex]$$
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
$$[/tex]
Since the tree is [tex]$186$[/tex] feet tall (opposite side) and the distance is [tex]$57$[/tex] feet (adjacent side), we compute:
[tex]$$
\tan \theta = \frac{186}{57} \approx 3.2632
$$[/tex]
To find the angle [tex]$\theta$[/tex], we take the inverse tangent (arctan) of this ratio:
[tex]$$
\theta = \arctan(3.2632)
$$[/tex]
Evaluating [tex]$\arctan(3.2632)$[/tex], we obtain:
[tex]$$
\theta \approx 1.2734 \text{ radians}
$$[/tex]
Since the answer needs to be in degrees, we convert from radians to degrees using the conversion factor [tex]$180^\circ/\pi$[/tex]:
[tex]$$
\theta \approx 1.2734 \times \frac{180^\circ}{\pi} \approx 72.9622^\circ
$$[/tex]
Finally, rounding the result to the nearest tenth of a degree gives:
[tex]$$
\theta \approx 73.0^\circ
$$[/tex]
Thus, the angle of elevation from level ground measured from [tex]$57$[/tex] feet away is approximately [tex]$\boxed{73.0^\circ}$[/tex].
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