Question: A 13 foot string of lights will be attached to the top of a 12 foot pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?
1. **State the problem:** We have a pole that is 12 feet tall and a string of lights that is 13 feet long. The string is attached at the top of the pole and anchored to the ground. We need to find the distance from the base of the pole to the point where the string is anchored.
2. **Identify the shape and formula:** This forms a right triangle where:
- The pole is one leg of length $12$ feet.
- The distance from the base to the anchor point is the other leg, which we call $x$.
- The string is the hypotenuse of length $13$ feet.
We use the Pythagorean theorem:
$$x^2 + 12^2 = 13^2$$
3. **Apply the Pythagorean theorem:**
$$x^2 + 144 = 169$$
4. **Isolate $x^2$:**
$$x^2 = 169 - 144$$
$$x^2 = 25$$
5. **Solve for $x$:**
$$x = \sqrt{25}$$
$$x = 5$$
6. **Interpret the result:** The string should be anchored 5 feet from the base of the pole.