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Zorluk: KolayAngles of Elevation, Depression, and Bearings

A surveyor stands at a point on level ground 50 m50\text{ m} away from the base of a vertical transmission tower. If the angle of elevation from the observer's position on the ground to the top of the tower is 45∘45^\circ, what is the height of the tower in meters?

Cevap: 50 meters

Cevap

The height of the transmission tower is 50 meters50\text{ meters}.
In a right-angled triangle, the tangent of the angle of elevation equals the ratio of the height (opposite side) to the horizontal distance (adjacent side). Since tan⁡(45∘)=1\tan(45^\circ) = 1, the height of the tower must be equal to the horizontal distance of 50 m50\text{ m}.

Adım Adım Çözüm

1
Formulate the trigonometric relationship using the right triangle formed by the observer, the base of the tower, and the top of the tower.
tan⁡(45∘)=h50\tan(45^\circ) = \frac{h}{50}, where hh is the height of the tower.
The tangent ratio relates the opposite side (height of tower) to the adjacent side (distance along level ground).
2
Substitute the value of tan⁡(45∘)=1\tan(45^\circ) = 1 and solve for hh.
h=50×1=50 mh = 50 \times 1 = 50\text{ m}.
Multiplying the adjacent side length by tan⁡(45∘)\tan(45^\circ) yields the exact height.

Anahtar Kavram

Angle of elevation using basic right-triangle trigonometry
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Konular

Angles of Elevation, Depression, and Bearings

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