Question: A 15 m long ladder is placed against a tree. The top of the ladder reaches a point 14 m up the tree. How far away is the base of the ladder from the base of the tree? Give your answer to 1 d.p.
1. **State the problem:** We have a ladder leaning against a tree forming a right-angled triangle. The ladder is the hypotenuse with length $15$ m, the height up the tree is $14$ m, and we need to find the horizontal distance from the base of the ladder to the tree.
2. **Formula used:** We use the Pythagorean theorem for right triangles:
$$a^2 + b^2 = c^2$$
where $c$ is the hypotenuse, and $a$ and $b$ are the other two sides.
3. **Assign values:** Let $x$ be the horizontal distance from the base of the ladder to the tree. Then:
$$x^2 + 14^2 = 15^2$$
4. **Calculate:**
$$x^2 + 196 = 225$$
5. **Isolate $x^2$:**
$$x^2 = 225 - 196$$
$$x^2 = 29$$
6. **Find $x$ by taking the square root:**
$$x = \sqrt{29}$$
7. **Calculate the numerical value:**
$$x \approx 5.385$$
8. **Round to 1 decimal place:**
$$x \approx 5.4$$
**Final answer:** The base of the ladder is approximately $5.4$ m away from the base of the tree.