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The question in the image provides the following information:

- A 44.0-meter guy wire is attached to the top of a 32.1-meter antenna.
- The wire is connected to a point on the ground.
  
We are tasked with finding:
1. How far the point on the ground is from the base of the antenna.
2. The angle that the guy wire makes with the ground.

### Step 1: Solving for the distance to the point on the ground (adjacent side)

We can treat this as a right triangle:
- The guy wire is the hypotenuse (44.0 m).
- The height of the antenna is one leg of the triangle (32.1 m).

Using the Pythagorean theorem:
\[
\text{(hypotenuse)}^2 = \text{(opposite)}^2 + \text{(adjacent)}^2
\]
Let \( d \) be the distance from the base of the antenna to the point on the ground (adjacent side):
\[
44.0^2 = 32.1^2 + d^2
\]
We can solve for \( d \).

### Step 2: Solving for the angle

The angle \( \theta \) between the guy wire and the ground can be found using trigonometric ratios. Specifically, we can use the sine function, as we know the opposite side (height of the antenna) and the hypotenuse (guy wire length):
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{32.1}{44.0}
\]
Then solve for \( \theta \) using the inverse sine function:
\[
\theta = \sin^{-1}\left(\frac{32.1}{44.0}\right)
\]

I'll calculate both values now.1. The point on the ground is approximately **30.1 meters** away from the base of the antenna.
2. The angle the guy wire makes with the ground is approximately **46.8°**.

Would you like more details on how these values were calculated or have any other questions?

Here are 5 related questions to expand on this concept:
1. How would the distance to the point on the ground change if the guy wire length increased?
2. What if the height of the antenna was shorter—how would the angle change?
3. Can you solve for the length of the guy wire if the angle and distance to the base are known?
4. How would you use cosine instead of sine to find the angle in this scenario?
5. How would changing the guy wire's tension affect its attachment to the antenna?

**Tip:** In any right triangle, you can use different trigonometric ratios (sine, cosine, tangent) depending on the known sides to find missing angles or distances.

A 44.0-meter guy wire is attached to the top of a 32.1-meter antenna and to a point on the ground. How far is the point on the ground from the base of the antenna, and what angle does the guy wire make with the ground?

Learn how to calculate the distance from the base of a 32.1-meter antenna to the point on the ground where a 44.0-meter guy wire is attached, using the Pythagorean theorem. The solution also includes finding the angle the guy wire makes with the ground.

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