Certainly! Let's explore the problem step-by-step to find the lengths of the guy wires.
1. Understanding the Problem: We have an 18-meter tall pole, and guy wires are attached to a point [tex]\(\frac{2}{3}\)[/tex] of the way up the pole. The wires are anchored to the ground 5 meters away from the base of the pole.
2. Identify the Height Where Guy Wires are Attached:
- The total height of the pole is 18 meters.
- The point where the wires are attached is [tex]\(\frac{2}{3}\)[/tex] of the height of the pole.
- Thus, the height where the guy wires are attached is:
[tex]\[
\text{Height of attachment} = 18 \times \frac{2}{3} = 12 \text{ meters}
\][/tex]
3. Identify the Distance from the Base of the Pole:
- The wires are attached to a point on the ground 5 meters away from the base of the pole.
4. Using the Pythagorean Theorem:
- We have a right triangle where:
- One leg is the height where the wires are attached (12 meters).
- The other leg is the distance from the base of the pole to where the wires are anchored on the ground (5 meters).
- The hypotenuse is the length of the guy wires.
- According to the Pythagorean Theorem:
[tex]\[
\text{Length of guy wire}^2 = (\text{Height of attachment})^2 + (\text{Distance from base})^2
\][/tex]
[tex]\[
\text{Length of guy wire}^2 = (12)^2 + (5)^2
\][/tex]
[tex]\[
\text{Length of guy wire}^2 = 144 + 25
\][/tex]
[tex]\[
\text{Length of guy wire}^2 = 169
\][/tex]
[tex]\[
\text{Length of guy wire} = \sqrt{169} = 13 \text{ meters}
\][/tex]
5. Conclusion:
- The height where the guy wires are attached is 12 meters.
- The length of each guy wire is 13 meters.
So, the lengths of the guy wires are 13 meters each.
