https://file.aiquickdraw.com/mathbot/file/1729810303387phb9dytq.jpg

To solve this problem, we need to determine the distance between the points where the pairs of cables are attached to the radio tower. Here's the breakdown of the approach:

### Part (a) - **Steps to Solve the Problem:**

1. **Understand the setup:**
   - The radio tower is supported by two sets of cables on each side.
   - The cables are anchored 10 meters away from the base of the tower.
   - There are two different angles of inclination: 30° for the shorter cables and 60° for the longer cables.

2. **Use Trigonometry:**
   - For each cable, you can use the tangent function to determine the height of the tower where each cable is attached. The tangent of an angle in a right triangle is defined as:
     \[
     \text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}}
     \]
     Where:
     - \(\theta\) is the angle of inclination.
     - "Opposite" refers to the height of the tower segment where the cable is attached.
     - "Adjacent" is the distance from the base of the tower to the anchor point (10 m in this case).

3. **Calculate the height of each cable attachment point:**
   - For the shorter cable: \( \tan(30^\circ) = \frac{\text{height1}}{10} \)
   - For the longer cable: \( \tan(60^\circ) = \frac{\text{height2}}{10} \)
   
   Rearranging the formulas to solve for height1 and height2 gives:
   - \( \text{height1} = 10 \times \tan(30^\circ) \)
   - \( \text{height2} = 10 \times \tan(60^\circ) \)

4. **Determine the vertical distance between the points where the cables are attached:**
   - The difference between the two heights will give the distance between the attachment points.

### Part (b) - **Carrying Out the Plan:**

1. Calculate the heights:
   - \( \text{height1} = 10 \times \tan(30^\circ) = 10 \times 0.5774 \approx 5.77 \) meters.
   - \( \text{height2} = 10 \times \tan(60^\circ) = 10 \times 1.732 \approx 17.32 \) meters.

2. Find the difference between these heights:
   \[
   \text{Distance between attachment points} = \text{height2} - \text{height1} = 17.32 - 5.77 = 11.55 \text{ meters}
   \]

Thus, the distance between the points where the cables are attached to the tower is approximately **11.6 meters**, rounded to the nearest tenth of a meter.

Would you like further details or have any questions? 

Here are 5 related questions:
1. How does the tangent function relate to the angles and distances in this problem?
2. What other trigonometric functions could be used in similar problems?
3. How can this method be adapted if the distance from the tower to the anchor points changes?
4. What is the effect of different angles of inclination on the height of cable attachment points?
5. Could you solve this problem using the sine or cosine rules instead of tangent?

**Tip:** Always ensure your calculator is set to degrees when solving problems involving angles in degrees, like this one.

A radio tower is supported by four cables, as shown. The cables are anchored to the ground on each side at points 10 m from the base of the tower. The angle of inclination of the shorter cables is 30°. The angle of inclination of the longer cables is 60°. The installers must determine the distance between the points where the pairs of cables are attached to the tower. a) Explain the steps to solve this problem. b) Carry out your plan. Round your answer to the nearest tenth of a metre.

Learn how to determine the distance between points where cables are attached to a radio tower using trigonometry. With angles of inclination of 30° and 60°, use the tangent function to calculate the attachment heights and find the vertical distance. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation