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The problem asks to find the slant height of a cone, given:

- Diameter of the base: 26 feet
- Height of the cone: 14 feet

### Step-by-Step Solution:

1. **Interpret the given information**:
   - The base of the cone is circular, with a diameter of 26 feet. The radius of the base is half of the diameter:
     \[
     r = \frac{26}{2} = 13 \text{ feet}
     \]
   - The height of the cone is the perpendicular distance from the base to the tip, which is given as 14 feet.
   
2. **Use the Pythagorean theorem**:
   - If we split the cone lengthwise, we form a right triangle, where:
     - The height of the cone is one leg: \( h = 14 \text{ feet} \)
     - The radius is the other leg: \( r = 13 \text{ feet} \)
     - The slant height \( s \) is the hypotenuse, which we need to find.
   
   By the Pythagorean theorem:
   \[
   s^2 = r^2 + h^2
   \]
   Substituting the values:
   \[
   s^2 = 13^2 + 14^2 = 169 + 196 = 365
   \]
   Taking the square root of both sides:
   \[
   s = \sqrt{365} \approx 19.10 \text{ feet}
   \]

### Final Answer:
The slant height of the pile is approximately **19.10 feet**.

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How would the slant height change if the diameter were doubled but the height stayed the same?
2. What is the surface area of the cone based on this slant height?
3. How do you calculate the volume of a cone, and what would it be for this cone?
4. What other real-world applications involve calculating slant height?
5. If the cone was tilted slightly, how would that affect the slant height calculation?

**Tip:** When working with cones, always remember the relationship between the height, radius, and slant height using the Pythagorean theorem.

Road salt is stored in a cone-shaped building. The base of the cone has a diameter of 26 ft and the height of the cone is 14 ft. What is the slant height of the pile?

Learn how to calculate the slant height of a cone with a base diameter of 26 ft and a height of 14 ft. This geometry problem involves the Pythagorean theorem to find the slant height, which is approximately 19.10 feet. Ideal for students in Grades 9-12.

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