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Zorluk: OrtaRight Triangles and the Pythagorean Theorem

A drone is hovering directly above a point on the ground. A surveyor's rangefinder, positioned 1.61.6 meters above the ground, measures the angle of elevation to the drone to be 60∘60^\circ. The rangefinder is located at a horizontal distance of 12312\sqrt{3} meters from the point directly beneath the drone. What is the height, in meters, of the drone above the ground?

  1. A
    13.613.6
  2. 37.637.6Cevap
  3. C
    36.036.0
  4. D
    25.625.6

Cevap

The height of the drone above the ground is 37.637.6 meters.
To find the height of the drone, we first represent the situation with a right triangle. The horizontal distance from the point directly below the drone to the rangefinder is 12312\sqrt{3} meters. The angle of elevation is 60∘60^\circ. In this right triangle, the side opposite the 60∘60^\circ angle represents the vertical height of the drone above the level of the rangefinder, which we can call hh. Using the tangent ratio, we have tan⁡(60∘)=h123\tan(60^\circ) = \frac{h}{12\sqrt{3}}. Since tan⁡(60∘)=3\tan(60^\circ) = \sqrt{3}, we find h=123×3=36h = 12\sqrt{3} \times \sqrt{3} = 36 meters. Finally, we add the height of the rangefinder above the ground to find the total height of the drone: 36+1.6=37.636 + 1.6 = 37.6 meters.

Adım Adım Çözüm

1
Identify the right triangle components from the given scenario.
A right triangle is formed where the horizontal leg (adjacent to the 60∘60^\circ angle) has a length of 12312\sqrt{3} meters, and the vertical leg (opposite the 60∘60^\circ angle) represents the height hh of the drone above the level of the rangefinder.
This sets up the geometric model to use trigonometric ratios.
2
Use the tangent trigonometric ratio to find the vertical height hh above the rangefinder level.
tan⁡(60∘)=h123⇒3=h123⇒h=123×3=36\tan(60^\circ) = \frac{h}{12\sqrt{3}} \Rightarrow \sqrt{3} = \frac{h}{12\sqrt{3}} \Rightarrow h = 12\sqrt{3} \times \sqrt{3} = 36 meters.
The tangent function relates the opposite side to the adjacent side in a right triangle.
3
Calculate the total height of the drone above the ground by adding the height of the rangefinder.
Total Height=36+1.6=37.6\text{Total Height} = 36 + 1.6 = 37.6 meters.
The rangefinder is positioned 1.61.6 meters above the ground, so the drone's height relative to the ground is the sum of its height above the rangefinder and the rangefinder's height.

Anahtar Kavram

Solving right triangles using trigonometric ratios (specifically tangent) or special 30-60-90 right triangle properties, and accounting for the height of the observer.
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Konular

Right Triangles and the Pythagorean Theorem

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