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A pole 12 feet tall is used to support a guy wire for a tower, which runs from the tower to a metal stake in the ground. After placing the pole, Travis measures the distance from the pole to the stake and from the pole to the tower, as shown in the diagram below. Find the length of the guy wire, to the nearest foot.


To solve this problem and find the length of the guy wire, we can approach it using the Pythagorean theorem, since we are dealing with a right triangle formed by the pole, the distance on the ground, and the guy wire.

### Step-by-step:

1. **Determine the distances:**
   - The distance between the tower and the pole is \( 23 \, \text{ft} \).
   - The distance between the pole and the metal stake is \( 5 \, \text{ft} \).
   - So, the total horizontal distance (base of the triangle) from the tower to the metal stake is:
   \[
   \text{Total horizontal distance} = 23 \, \text{ft} + 5 \, \text{ft} = 28 \, \text{ft}.
   \]
   
2. **Height of the pole:**
   - The height of the pole is \( 12 \, \text{ft} \).

3. **Use the Pythagorean theorem:**
   The guy wire forms the hypotenuse of the right triangle, with the height of the pole as one leg and the total horizontal distance as the other leg.
   The Pythagorean theorem is given by:
   \[
   a^2 + b^2 = c^2,
   \]
   where:
   - \( a = 12 \, \text{ft} \) (the height),
   - \( b = 28 \, \text{ft} \) (the horizontal distance),
   - \( c \) is the length of the guy wire (the hypotenuse).

4. **Solve for the guy wire length:**
   \[
   c^2 = 12^2 + 28^2 = 144 + 784 = 928.
   \]
   Taking the square root of both sides:
   \[
   c = \sqrt{928} \approx 30.46 \, \text{ft}.
   \]

Therefore, the length of the guy wire is approximately **30 feet**.

Would you like more details or clarifications? Here are some related questions:

1. How would the length of the guy wire change if the distance to the stake was different?
2. What happens to the height of the pole if the guy wire length changes?
3. Can we use trigonometry instead of the Pythagorean theorem in this situation?
4. How do we ensure the pole stays stable with the guy wire attached?
5. What other factors affect the strength of a guy wire?

**Tip:** When using the Pythagorean theorem, ensure that all measurements are in the same units before solving for the unknown value.

A pole 12 feet tall is used to support a guy wire for a tower, which runs from the tower to a metal stake in the ground. After placing the pole, Travis measures the distance from the pole to the stake and from the pole to the tower, as shown in the diagram below. Find the length of the guy wire, to the nearest foot.

This problem demonstrates how to calculate the length of a guy wire using the Pythagorean theorem. A 12-foot pole supports a wire extending to a metal stake, with a total horizontal distance of 28 feet from the tower to the stake. Learn how to apply the formula a^2 + b^2 = c^2 to find the wire length.

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