To find the volume of the cone, we use the formula for the volume of a cone:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( \pi \) is approximately 3.14,
- \( r \) is the radius,
- \( h \) is the height.
Given for the cone:
- \( r = 3 \) feet,
- \( h = 11 \) feet.
Plugging in the values:
\[ V = \frac{1}{3} \times 3.14 \times (3^2) \times 11 \]
First, calculate \( (3^2) \):
\[ (3^2) = 9 \]
Now move forward with the volume computation:
\[ V = \frac{1}{3} \times 3.14 \times 9 \times 11 \]
Calculate \( 9 \times 11 \):
\[ 9 \times 11 = 99 \]
Then:
\[ V = \frac{1}{3} \times 3.14 \times 99 \]
Now calculate \( 3.14 \times 99 \):
\[ 3.14 \times 99 \approx 310.86 \]
Now divide by 3:
\[ V = \frac{310.86}{3} \approx 103.62 \]
Thus, the volume of the cone is approximately 103.62 cubic feet.
Next, we find the radius of the cylinder using its volume formula. The volume of a cylinder is given by:
\[ V = \pi r^2 h \]
Given:
- \( V = 310.86 \) cubic feet,
- \( h = 11 \) feet.
Using the formula, we can rearrange it to find \( r \):
\[ r^2 = \frac{V}{\pi h} \]
Substituting the known values:
\[ r^2 = \frac{310.86}{3.14 \times 11} \]
Calculate \( 3.14 \times 11 = 34.54 \):
\[ r^2 = \frac{310.86}{34.54} \approx 9.00 \]
Taking the square root:
\[ r = \sqrt{9.00} = 3.00 \]
So, the radius of the cylinder is 3.00 feet.
Lastly, to find the ratio of the volume of the cone to the volume of the cylinder, we have:
\[ \text{Ratio} = \frac{\text{Volume of cone}}{\text{Volume of cylinder}} = \frac{103.62}{310.86} \]
Calculating the ratio:
\[ \text{Ratio} \approx 0.3333 \]
Thus, the ratio can be represented as:
\[ 1 : 3 \]
So the final answers are:
The volume of the cone is 103.62 cubic feet.
The radius of the cylinder is 3.00 feet.
The ratio of the volume of the cone to the volume of the cylinder is 1:3.