To find the angle that the guy wire makes with the ground, we can use trigonometry. Specifically, the cosine function is suitable because we know the adjacent side (the distance from the base of the tower to where the wire touches the ground) and the hypotenuse (the length of the wire).

Step 1: Define the cosine function

The cosine of the angle $\theta$ is given by: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Here:

• Adjacent side = 510 feet
• Hypotenuse = 1690 feet

Step 2: Solve for $\cos(\theta)$

$\cos(\theta) = \frac{510}{1690} \approx 0.3018$

Step 3: Find the angle $\theta$

Take the inverse cosine ($\cos^{-1}$) to find $\theta$: $\theta = \cos^{-1}(0.3018)$

Using a calculator: $\theta \approx 72.47^\circ$

Final Answer:

The angle the wire makes with the ground is approximately 72.47°.

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Would you like a step-by-step breakdown of using the cosine function or have any other questions?

5 Related Questions:

1. How do you calculate the height of the tower in this problem?
2. Can sine or tangent be used instead of cosine here?
3. What is the role of the Pythagorean theorem in verifying this solution?
4. How would the angle change if the length of the guy wire were increased?
5. How can you determine the angle if only the height of the tower is given?

Tip:

When working with angles in trigonometry, always double-check whether your calculator is set to degrees or radians to ensure accurate results.