Question 1170415
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.