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The problem involves determining the length of a cable that runs from the top of a 29-foot tall vertical pole on a hill to a point 78 feet downhill. The hill makes an angle of \(16^\circ\) with the horizontal, as shown in the diagram.

This forms a right triangle where:
- The height of the pole is 29 feet (opposite side),
- The horizontal distance downhill is 78 feet (adjacent side),
- The length of the cable is the hypotenuse (which we are solving for).

We can use the **Pythagorean Theorem** to calculate the hypotenuse:

\[
\text{Hypotenuse}^2 = \text{Height}^2 + \text{Base}^2
\]

The base isn't just the 78 feet because the hill is inclined, so we need to adjust for that using trigonometry.

1. Find the true horizontal distance \(d_{\text{horizontal}}\) using:
\[
d_{\text{horizontal}} = 78 \cdot \cos(16^\circ)
\]

2. Now, apply the Pythagorean theorem to find the cable length \(L\):
\[
L = \sqrt{29^2 + d_{\text{horizontal}}^2}
\]

Let me calculate this for you:The approximate length of the cable needed is **80.39 feet**. However, the closest given answer choices are:

- 75.35 feet
- 88.4 feet
- 82.88 feet
- 90.4 feet

None of these match exactly, but based on the calculation, the correct answer should be around **80.39 feet**, which could suggest an issue with rounding or available answer choices in this question.

Do you have any questions, or would you like more details on how this was calculated?

Here are some follow-up questions:
1. How would the result change if the angle of the hill were steeper?
2. What formula would you use if the pole wasn’t perfectly vertical?
3. How can we verify this calculation in a different way?
4. How do you apply trigonometry to solve real-world slope problems?
5. What role does the angle of elevation play in other geometric problems?

**Tip:** Remember that the Pythagorean theorem is only valid in right triangles, so it’s important to check angles carefully in word problems.

A vertical pole 29 feet tall stands on a hill that makes an angle of 16 degrees with the horizontal. Determine the approximate length of a cable that would be needed to reach from the top of the pole to a point 78 feet downhill from the base of the pole.

This problem uses the Pythagorean Theorem and trigonometry to determine the length of a cable from a 29-foot tall pole to a point 78 feet downhill, with a hill slope of 16 degrees. A perfect problem for practicing right triangle calculations for Grades 9-11.

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