https://file.aiquickdraw.com/mathbot/file/1727172947998lsdwdoqt.jpg

The problem in the image asks for the minimum length of the cable needed to connect two terrain points, A and B, based on the diagram provided. 

### Known information:
1. The height at point A is 1.5 meters.
2. The height at point B is 2.75 meters.
3. The angle between the horizontal and the line from point A to B is \( 14.5^\circ \).

### Objective:
We need to find the horizontal distance between points A and B and then calculate the cable length using trigonometry.

### Steps:
1. **Height Difference**:
   - The difference in heights between points A and B is:
     \[
     \Delta h = 2.75 - 1.5 = 1.25 \, \text{meters}
     \]

2. **Trigonometric Setup**:
   - We can use the tangent function since the angle and the height difference are given:
     \[
     \tan(\theta) = \frac{\Delta h}{d_{\text{horizontal}}}
     \]
     where \( \theta = 14.5^\circ \) and \( d_{\text{horizontal}} \) is the horizontal distance between points A and B.

3. **Calculate Horizontal Distance**:
   Rearranging the tangent equation to solve for the horizontal distance:
     \[
     d_{\text{horizontal}} = \frac{\Delta h}{\tan(14.5^\circ)}
     \]

4. **Find Cable Length**:
   - Once the horizontal distance is known, the cable length can be calculated using the Pythagorean theorem:
     \[
     L_{\text{cable}} = \sqrt{(d_{\text{horizontal}})^2 + (1.25)^2}
     \]

Let me calculate these values for you.The horizontal distance between points A and B is approximately \( 4.83 \, \text{m} \). 

The minimum length of the cable required to connect points A and B is approximately \( 4.99 \, \text{m} \).

Would you like more details or explanations? Here are five related questions that might interest you:

1. How does the angle of elevation affect the required cable length?
2. What is the significance of using the Pythagorean theorem in this context?
3. How would the cable length change if the height difference increased?
4. Can we calculate the horizontal distance using sine or cosine instead of tangent?
5. How does changing the angle of elevation impact the design of real-world structures?

**Tip**: In problems involving angles and heights, always try to relate given heights with horizontal distances through trigonometric functions like sine, cosine, or tangent.

Wie lang muss das Kabel mindestens sein, das zwischen den Gelände­punkten A und B im Boden verlegt werden soll?

Learn how to calculate the minimum cable length needed to connect two terrain points A and B using trigonometric principles. This problem involves applying the tangent function and the Pythagorean theorem to find the horizontal distance and the total cable length. Suitable for Grades 9-12 students.

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