To find the difference in volume between the two cones, we first need to calculate the volume of each cone using the formula for the volume of a cone:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
For the first cone (radius = 12 feet, height = 21 feet):
\[ V_1 = \frac{1}{3} \pi (12)^2 (21) \]
Calculating this step by step:
- Calculate \( (12)^2 = 144 \).
- Then, calculate \( V_1 = \frac{1}{3} \times 3.14 \times 144 \times 21 \).
Now, calculate \( 144 \times 21 \):
\[ 144 \times 21 = 3024 \]
Now substitute back into the volume formula:
\[ V_1 = \frac{1}{3} \times 3.14 \times 3024 \]
Calculate \( 3.14 \times 3024 \):
\[ 3.14 \times 3024 \approx 9485.76 \]
Now, finally calculate \( V_1 \):
\[ V_1 = \frac{1}{3} \times 9485.76 \approx 3161.92 \text{ cubic feet} \]
For the second cone (radius = 6 feet, height = 21 feet):
\[ V_2 = \frac{1}{3} \pi (6)^2 (21) \]
Following similar steps:
- Calculate \( (6)^2 = 36 \).
- Calculate \( V_2 = \frac{1}{3} \times 3.14 \times 36 \times 21 \).
Now calculate \( 36 \times 21 \):
\[ 36 \times 21 = 756 \]
Now substitute into the volume formula:
\[ V_2 = \frac{1}{3} \times 3.14 \times 756 \]
Calculate \( 3.14 \times 756 \):
\[ 3.14 \times 756 \approx 2377.44 \]
Now calculate \( V_2 \):
\[ V_2 = \frac{1}{3} \times 2377.44 \approx 792.48 \text{ cubic feet} \]
Now, calculate the difference in volumes:
\[ \text{Difference} = V_1 - V_2 = 3161.92 - 792.48 \approx 2369.44 \text{ cubic feet} \]
Thus, the difference in the volumes of the cones is approximately:
\[ \boxed{2369.44} \text{ cubic feet} \]