Two vertical poles, 3m and 7m long are on the same straight line with a point P on the ground. The shorter pole is 20m from P and the longer pole. The angle of elevation of the top T of the longer pole from the top R of the shorter one is 30degree, calculate i) /RT/ ii) the horizontal distance from P to the longer pole, iii) the angle of elevation of T from P.
i) 20/RT = cos30°
ii) 20+20
iii) tanθ = 7/40
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To solve this problem, let's name the points as follows:
- Point R: Top of the shorter pole
- Point T: Top of the longer pole
- Point P: Point on the ground
Given information:
- Height of the shorter pole, /RP/: 3m
- Height of the longer pole, /RT/: 7m
- Distance from P to both poles: 20m
- Angle of elevation of T from R: 30 degrees
To find the solutions, let's solve each part step by step:
i) To find /RT/ (height of the longer pole):
We can use the tangent function. In a right-angled triangle with angle θ, opposite side O and adjacent side A:
tangent(θ) = O / A
In this case, O = /RT/ and A = 20m.
So, tangent(30 degrees) = /RT/ / 20m
Solving for /RT/:
/RT/ = tangent(30 degrees) × 20m
Using a calculator:
/RT/ = 0.577 × 20m
/RT/ = 11.54m (approximately)
ii) To find the horizontal distance from P to the longer pole:
We can use the cosine function. In a right-angled triangle with angle θ, adjacent side A (horizontal distance), and hypotenuse H:
cosine(θ) = A / H
In this case, θ = 30 degrees and H = 20m.
So, cosine(30 degrees) = A / 20m
Solving for A:
A = 20m × cosine(30 degrees)
Using a calculator:
A ≈ 20m × 0.866
A ≈ 17.32m
Therefore, the horizontal distance from P to the longer pole is approximately 17.32m.
iii) To find the angle of elevation of T from P:
We can use the tangent function. In a right-angled triangle with angle θ, opposite side O (height of the longer pole), and adjacent side A (horizontal distance):
tangent(θ) = O / A
In this case, O = 7m and A = 17.32m.
So, tangent(θ) = 7m / 17.32m
Solving for θ:
θ = arc-tangent(7m / 17.32m)
Using a calculator:
θ ≈ 22.76 degrees
Therefore, the angle of elevation of T from P is approximately 22.76 degrees.
To solve this problem, we can use trigonometry. Let's go step by step:
i) To find /RT/, we can use the sine function.
Let's consider the shorter pole as pole A and the longer pole as pole B. We are given that the angle of elevation of the top T of pole B from the top R of pole A is 30 degrees.
/RT/ represents the height of pole B above pole A, so we need to find the length of line segment RT.
Using the sine function, sin(30°) = opposite/hypotenuse.
The length of the shorter pole is the adjacent side, and we need to find the opposite side, so let's call it x.
sin(30°) = x / 3
x = 3 * sin(30°)
x ≈ 1.5 meters
Therefore, /RT/ ≈ 1.5 meters.
ii) To find the horizontal distance from P to pole B, let's call it d.
We are given that the shorter pole is 20 meters from point P and pole B. We need to find the value of d.
We can use the tangent function to find d.
tan(30°) = opposite/adjacent.
The opposite side is the distance from the shorter pole to point P, which is 20 meters.
The adjacent side is the distance we need to find, d.
tan(30°) = 20 / d
d = 20 / tan(30°)
d ≈ 34.64 meters
Therefore, the horizontal distance from P to pole B is approximately 34.64 meters.
iii) To find the angle of elevation of T from P, let's call it θ.
We need to use the tangent function again.
tan(θ) = opposite/adjacent
The opposite side is the height of pole B above pole A, which we found in part i) to be approximately 1.5 meters.
The adjacent side is the horizontal distance from P to pole B, which we found in part ii) to be approximately 34.64 meters.
tan(θ) = 1.5 / 34.64
θ ≈ 2.5 degrees
Therefore, the angle of elevation of T from P is approximately 2.5 degrees.