To determine the total volume of the cone, we use the formula for the volume of a cone:

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

Where:

• $r$ is the radius of the cone,
• $h$ is the height of the cone,
• $\pi$ is the constant pi.

Given:

• Radius $r = 9n^2$,
• Height $h = 9$.

Step-by-Step Solution:

1. Substitute the given values into the volume formula: $V = \frac{1}{3} \pi (9n^2)^2 (9)$

2. Simplify $(9n^2)^2$: $(9n^2)^2 = 81n^4$

3. Substitute back into the formula: $V = \frac{1}{3} \pi (81n^4)(9)$

4. Multiply the constants: $81 \times 9 = 729$ So: $V = \frac{1}{3} \pi (729n^4)$

5. Simplify: $V = 243\pi n^4$

Final Answer:

The total volume of the cone is: $\boxed{243\pi n^4}$

Do you need further clarification or have more questions about this? Let me know!

Related Questions:

1. How do you derive the formula for the volume of a cone?
2. What is the surface area of this cone in terms of $\pi$?
3. If the cone's height were doubled, what would the new volume be?
4. What would the volume be if the radius were halved?
5. Can you write the volume in decimal form for specific $n$ values?

Tip:

Always ensure you square the radius before multiplying by the height when using the cone volume formula!