Question 1170415: a man sitting on 22.5 m high storey building measures the angles of elevation of a telecom mast directly ahead of him as 43°. He measures the angle of elevation of the mast when he is 52m in the front of the building on the horizontal ground as 70°
A calculate the height of the mast
The distance between the building and the mast.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**Diagram:**
Imagine the following:
* **B:** The base of the building.
* **T:** The top of the building.
* **M:** The top of the telecom mast.
* **H:** The base of the telecom mast.
* **P1:** The man's first position on the building.
* **P2:** The man's second position on the ground.
**Given:**
* Building height (BT) = 22.5 m
* Angle of elevation from P1 to M = 43°
* Distance P2B = 52m
* Angle of elevation from P2 to M = 70°
**Let's define:**
* Height of the mast (HM) = h
* Distance between the building and the mast (BH) = x
**Step 1: Analyzing the first position (P1)**
* The man is on the top of the building, so TP1 is horizontal.
* Let's draw a horizontal line from T to a point on HM, let's call it A.
* Then, TA = BH = x, and AM = h - 22.5.
* In triangle TAM, we have:
* tan(43°) = AM / TA = (h - 22.5) / x
* x = (h - 22.5) / tan(43°)
**Step 2: Analyzing the second position (P2)**
* In triangle HMP2, we have:
* tan(70°) = HM / HP2 = h / x
* x = h / tan(70°)
**Step 3: Equating the two expressions for x**
* Since both expressions equal x, we can set them equal to each other:
* (h - 22.5) / tan(43°) = h / tan(70°)
* (h - 22.5) * tan(70°) = h * tan(43°)
* h * tan(70°) - 22.5 * tan(70°) = h * tan(43°)
* h * tan(70°) - h * tan(43°) = 22.5 * tan(70°)
* h (tan(70°) - tan(43°)) = 22.5 * tan(70°)
* h = (22.5 * tan(70°)) / (tan(70°) - tan(43°))
**Step 4: Calculating h (height of the mast)**
* tan(70°) ≈ 2.7475
* tan(43°) ≈ 0.9325
* h = (22.5 * 2.7475) / (2.7475 - 0.9325)
* h = 61.81875 / 1.815
* h ≈ 34.06 m
**Step 5: Calculating x (distance between the building and the mast)**
* x = h / tan(70°)
* x = 34.06 / 2.7475
* x ≈ 12.40 m
**Therefore:**
* The height of the mast is approximately 34.06 meters.
* The distance between the building and the mast is approximately 12.40 meters.
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