13. A ladder rests against the top of the perpendicular wall of a building and makes an angle of $$ 2 M $$ from the wall, calculate the height of the building' 14. An electric plylon stands on horizontal ground. At a point som from the base of the pylon, the angle of elevation of the top of the pylon is $$ 23^{\circ} $$. Calculate the height of the pylon

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---

### **Problem 13**

**Statement:**
*A ladder rests against the top of the perpendicular wall of a building and makes an angle of $$ 2M $$ from the wall. Calculate the height of the building.*

**Assumptions & Given Data:**
- Let’s interpret $$ 2M $$ as an angle measurement. For clarity, we'll assume $$ M $$ is in **degrees**. So, the ladder makes an angle of $$ 2M^\circ $$ with the wall.
- Let’s denote:
  - $$ \theta = 2M^\circ $$ (the angle between the ladder and the wall)
  - $$ L $$ = Length of the ladder
  - $$ H $$ = Height of the building (which we need to find)

**Diagram:**
```
          |
          |\
          | \
        H |  \ L
          |   \
          |____\
             D
```
- Here, $$ H $$ is the height of the building.
- $$ D $$ is the horizontal distance from the base of the wall to the foot of the ladder.

**Trigonometric Relationship:**
The ladder, wall, and ground form a right-angled triangle. We can use the **cosine** of the angle to relate $$ H $$ and $$ L $$:

$$
\cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{H}{L}
$$

**Solving for $$ H $$:**

$$
H = L \cdot \cos(\theta) = L \cdot \cos(2M^\circ)
$$

**Conclusion:**
The height of the building ($$ H $$) can be calculated using the length of the ladder ($$ L $$) and the angle $$ 2M^\circ $$ it makes with the wall:

$$
\boxed{H = L \cdot \cos(2M^\circ)}
$$

*Note:* To find a numerical value for $$ H $$, you'd need the specific length of the ladder ($$ L $$) and the value of $$ M $$.

---

### **Problem 14**

**Statement:**
*An electric pylon stands on horizontal ground. At a point some distance from the base of the pylon, the angle of elevation of the top of the pylon is $$ 23^\circ $$. Calculate the height of the pylon.*

**Assumptions & Given Data:**
- Let’s denote:
  - $$ \alpha = 23^\circ $$ (the angle of elevation)
  - $$ D $$ = Horizontal distance from the observation point to the base of the pylon
  - $$ H $$ = Height of the pylon (which we need to find)

**Diagram:**
```
          |
          |\
          | \
        H |  \ 
          |   \
          |____\
             D
```

**Trigonometric Relationship:**
The scenario forms a right-angled triangle where the pylon is the perpendicular side, the distance $$ D $$ is the base, and the line of sight is the hypotenuse. We can use the **tangent** of the angle to relate $$ H $$ and $$ D $$:

$$
\tan(\alpha) = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{H}{D}
$$

**Solving for $$ H $$:**

$$
H = D \cdot \tan(\alpha) = D \cdot \tan(23^\circ)
$$

**Conclusion:**
The height of the pylon ($$ H $$) can be calculated using the horizontal distance ($$ D $$) and the angle of elevation ($$ 23^\circ $$):

$$
\boxed{H = D \cdot \tan(23^\circ)}
$$

**Numerical Example:**
If, for instance, the distance $$ D $$ is **50 meters**, then:

$$
H = 50 \cdot \tan(23^\circ) \approx 50 \cdot 0.4245 \approx 21.23 \text{ meters}
$$

*Note:* Replace **50 meters** with the actual distance to compute the precise height.

---

**Summary:**

- **Problem 13:** The height of the building is determined by the length of the ladder and the angle it makes with the wall using the cosine function.
  
  $$
  H = L \cdot \cos(2M^\circ)
  $$

- **Problem 14:** The height of the pylon is determined by the horizontal distance from the observation point and the angle of elevation using the tangent function.

$$
  H = D \cdot \tan(23^\circ)
  $$

Ensure that you have the necessary measurements (ladder length $$ L $$ for Problem 13 and distance $$ D $$ for Problem 14) to compute the numerical values of the heights.
To find the height of the building in Problem 13, use the formula: $$ \text{Height} = \text{Ladder Length} \times \cos(2M^\circ) $$ For Problem 14, the height of the pylon is calculated as: $$ \text{Height} = \text{Distance from Pylon} \times \tan(23^\circ) $$ Make sure to have the ladder length and distance from the pylon to find the exact heights.

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