This problem involves finding the volume of a cone, which is calculated using the formula:

$V = \frac{1}{3} \pi r^2 h$

From the diagram:

• The slant height is $13 \, \text{cm}$.
• The height of the cone is $h = 12 \, \text{cm}$.
• We need to find the radius $r$ using the Pythagorean theorem, since the slant height, height, and radius form a right triangle.

Step 1: Apply the Pythagorean theorem

$r^2 + h^2 = (\text{slant height})^2$ Substituting the values: $r^2 + 12^2 = 13^2$ $r^2 + 144 = 169$ $r^2 = 169 - 144 = 25$ $r = \sqrt{25} = 5 \, \text{cm}$

Step 2: Calculate the volume

Now that we know $r = 5 \, \text{cm}$ and $h = 12 \, \text{cm}$, substitute into the cone volume formula: $V = \frac{1}{3} \pi r^2 h$ $V = \frac{1}{3} \pi (5^2)(12)$ $V = \frac{1}{3} \pi (25)(12)$ $V = \frac{1}{3} \pi (300)$ $V = 100 \pi \, \text{cm}^3$

Final Answer:

The exact volume of the cone is: $\boxed{100 \pi \, \text{cm}^3}$

Let me know if you need further clarification!

Questions for Expansion:

1. What would the volume be if the slant height was increased to 15 cm while keeping the height constant?
2. How does the formula for the volume of a cone differ from that of a cylinder?
3. Can we derive the formula for the volume of a cone using integration?
4. How would the volume change if the radius were doubled but the height remained the same?
5. How do we calculate the surface area of this cone?

Tip:

Always confirm the radius using the Pythagorean theorem when given the slant height and height of a cone.