The volume $V$ of a cone can be calculated with the formula:

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

where:

• $r$ is the radius of the base,
• $h$ is the height.

Step 1: Determine the Radius

Given the diameter is 12, we can find the radius by dividing by 2: $r = \frac{12}{2} = 6$

Step 2: Substitute Values into the Formula

Substitute $r = 6$ and $h = 19$: $V = \frac{1}{3} \pi (6)^2 \cdot 19$

Step 3: Simplify the Expression

1. Calculate $r^2$: $r^2 = 6^2 = 36$
2. Substitute back: $V = \frac{1}{3} \pi \cdot 36 \cdot 19$
3. Multiply $36$ and $19$: $V = \frac{1}{3} \pi \cdot 684$
4. Now divide by 3: $V = 228 \pi$

Final Answer

• Exact value: $V = 228 \pi$
• Approximate value (using $\pi \approx 3.14$): $V \approx 228 \times 3.14 = 715.92$

Thus, the volume of the cone is approximately 715.92 cubic units.

Would you like further details or have questions?

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Here are some questions that expand on this topic:

1. How would the volume change if the height were doubled?
2. What is the effect on the volume if the radius is tripled?
3. How do we derive the formula for the volume of a cone?
4. What are the similarities and differences in calculating the volumes of a cone and a cylinder?
5. How is the concept of volume applied in real-life problems involving cones?

Tip: Always keep track of units throughout calculations for clarity.