https://file.aiquickdraw.com/mathbot/file/1730951672152sp0uupdh.jpg

To solve this problem, let's break it down into steps. We have a right triangle where:

1. **The height of the flagpole (vertical side)** = \(10.6 \, \text{m}\)
2. **The distance from the base of the pole to the point on the ground (horizontal side)** = \(5.5 \, \text{m}\)
3. **The length of the cable (hypotenuse)** = \(12.3 \, \text{m}\)

We are asked to find:

1. **The angle between the cable and the ground**
2. **The angle between the cable and the pole**

### Step 1: Calculate the Angles

Let’s denote:
- \(\theta\) as the angle between the **cable and the ground**.
- \(\alpha\) as the angle between the **cable and the pole**.

#### 1. Finding \(\theta\) (Angle between the cable and the ground)

Using trigonometry, specifically the cosine rule, we have:
\[
\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{5.5}{12.3}
\]
Calculating \(\theta\):
\[
\theta = \cos^{-1}\left(\frac{5.5}{12.3}\right)
\]

#### 2. Finding \(\alpha\) (Angle between the cable and the pole)

Similarly, using the sine rule:
\[
\sin(\alpha) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{10.6}{12.3}
\]
Calculating \(\alpha\):
\[
\alpha = \sin^{-1}\left(\frac{10.6}{12.3}\right)
\]

After solving these, we round each result to the nearest degree.

Let me calculate these values.The calculated angles are:

1. The angle between the cable and the ground is **63°**.
2. The angle between the cable and the pole is **60°**.

Would you like further details or have any questions?

Here are five related questions for practice:

1. How would the angle calculations change if the cable were shorter?
2. Can you determine the height of the pole if only the cable length and angle with the ground are known?
3. How would you calculate the horizontal distance if only the pole's height and the angle between the pole and the ground were known?
4. What if the pole were leaning in a different direction? How would that affect the calculations?
5. How can the Pythagorean theorem be used here to verify the measurements?

**Tip:** In trigonometry, identifying the correct sides of a triangle relative to the angle you’re interested in is key to choosing the right function (sine, cosine, or tangent).

A 10.6 m tall flagpole has a 12.3 m long cable connecting its top to a spot on the ground 5.5 m from the base of the pole. The pole is leaning slightly. Find the angle between the cable and the ground, the cable and the pole, and the obtuse angle between the pole and the ground. The tenth is given after rounding to the nearest degree.

Solve a trigonometry problem involving a 10.6 m flagpole, a 12.3 m cable, and a ground distance of 5.5 m. Learn how to find the angles between the cable and the ground, the cable and the pole, and the obtuse angle between the pole and the ground. This problem is suitable for high school 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