To measure the height of an inaccessible cliff on the opposite side of a river, a surveyor makes the measurements shown in the figure ( one angle is 33.1 & the other i 69.4) . If distance d = 202 meters and Ð b = 50.0°, find the height of the cliff.
the diagram is a triangle then a flipped image of a triangle connected together.. if that makes sense - the one triangle is a right angle and the second mirror image contains the 2 angles 69.4 and 50.
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Using the Law of Sines, we can calculate the height of the cliff:
h = (202 m) * (sin(33.1°)) / (sin(50.0°))
h = (202 m) * (0.547) / (0.766)
h = 144.3 m
To find the height of the cliff, we can use trigonometry. Let's break down the problem step-by-step:
Step 1: Identify the relevant sides and angles in the given diagram.
In the right-angled triangle, we have:
- One angle: θ = 33.1°
- One side opposite to θ: d = 202 meters (the distance from the surveyor to the base of the cliff)
In the other triangle, we have:
- One angle: φ = 69.4°
- One angle: β = 50.0°
We need to find the height of the cliff represented by "h."
Step 2: Determine which trigonometric ratio is appropriate for this scenario.
Since we have the opposite side and the adjacent side, we'll use the tangent function, which is defined as:
tan(θ) = opposite/adjacent.
In this case, we'll use the tangent of θ to find h.
Step 3: Use trigonometry to find the height of the cliff.
We know that:
tan(θ) = h/d,
Rearranging the equation, we get:
h = tan(θ) * d.
Substituting the given values, we have:
h = tan(33.1°) * 202.
Calculating this expression, we find:
h ≈ 119.98 meters.
Therefore, the height of the cliff is approximately 119.98 meters.
To find the height of the cliff, we'll use the principles of trigonometry. Here's how we can proceed:
1. Label the triangle to keep track of the angles and sides. Let's call the right-angled triangle ABC, where A is the right angle, B is the 33.1° angle, and C is the 90° angle. The mirror image triangle on the opposite side of the river can be called A'B'C, which is a reflection of ABC.
2. Determine the unknown sides and angles. We're given that angle B = 33.1°, angle A'B'C = 69.4°, and angle B'AC = 50.0°. We also know that distance d = 202 meters.
3. Use the trigonometric ratio tangent (tan) to find the height of the cliff. Since we have the adjacent side (AB) and the angle opposite to the height (angle B), we can use the tangent:
tan(B) = height / d
Rearranging the equation, we find:
height = d * tan(B)
4. Substitute the given values into the equation and calculate the height:
height = 202 * tan(33.1°)
Using a calculator, the value of tan(33.1°) is approximately 0.649
height ≈ 202 * 0.649
height ≈ 131 meters
Therefore, the height of the cliff is approximately 131 meters.