This problem can be solved using the Pythagorean theorem. The utility pole, the ground, and the cable form a right triangle, where:

• The height of the pole is 8 m (one leg of the right triangle),
• The distance from the bottom of the pole to where the cable meets the ground is 4 m (the other leg),
• The cable is the hypotenuse (the longest side).

The Pythagorean theorem states that:

$a^2 + b^2 = c^2$

where:

• $a$ and $b$ are the lengths of the two legs, and
• $c$ is the length of the hypotenuse (the cable in this case).

In this case:

• $a = 8$ m (height of the pole),
• $b = 4$ m (distance from the pole to where the cable touches the ground).

Substitute these values into the Pythagorean theorem:

$8^2 + 4^2 = c^2$

$64 + 16 = c^2$

$80 = c^2$

Now, take the square root of both sides to solve for $c$:

$c = \sqrt{80} \approx 8.94 \, \text{m}$

So, the length of the cable is approximately 8.94 meters.

Would you like a more detailed breakdown or have any further questions? Here are some related questions to explore:

1. How can the Pythagorean theorem be used in real-world applications?
2. What happens if the distance from the pole to the cable increases or decreases?
3. How would you solve for the height of the pole if the cable length was different?
4. What is the relationship between the sides of a 3-4-5 right triangle, and how is it applied in this problem?
5. How do you determine if a triangle is a right triangle based on the side lengths?

Tip: Always double-check your units and ensure they are consistent when applying the Pythagorean theorem.