To find the volume of the cone in the image, we’ll use the volume formula for a cone:

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

where:

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

Given:

• $d = 12$ (diameter),
• $h = 13$ (height).

Step 1: Find the Radius

Since the diameter $d = 12$, the radius $r$ is half of the diameter: $r = \frac{d}{2} = \frac{12}{2} = 6$

Step 2: Substitute into the Volume Formula

Now, substitute $r = 6$ and $h = 13$ into the formula:

$V = \frac{1}{3} \pi (6)^2 (13)$

Calculating $6^2 = 36$: $V = \frac{1}{3} \pi \times 36 \times 13$

Then, $36 \times 13 = 468$: $V = \frac{1}{3} \pi \times 468$

Now, divide by 3: $V = 156 \pi$

Step 3: Approximate the Volume

To approximate $V$ to the nearest hundredth, use $\pi \approx 3.14159$: $V \approx 156 \times 3.14159 = 489.58$

Final Answer

• Exact Volume: $156\pi \, \text{ft}^3$
• Approximate Volume: $489.58 \, \text{ft}^3$

Would you like more details on any step?

Here are some related questions for practice:

1. How would the volume change if the height were doubled?
2. How would you find the volume if given only the surface area of the cone?
3. What is the volume of a cylinder with the same radius and height?
4. How does the volume of a cone compare to that of a sphere with the same diameter?
5. What would the radius need to be for a cone of the same height to have double this volume?

Tip: For cones, the volume is always one-third of a cylinder with the same base and height.