To find more information about the cone and the cylinder based on the given dimensions, we can start by calculating the volume of the cone since we have its radius and height.
Volume of the Cone
The formula for the volume of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given:
- \( r = 3 \) feet,
- \( h = 11 \) feet.
Substituting these values into the formula:
\[ V = \frac{1}{3} \pi (3^2)(11) \]
Calculating \( 3^2 \):
\[ 3^2 = 9 \]
Now, substituting that in:
\[ V = \frac{1}{3} \pi (9)(11) = \frac{1}{3} \pi (99) = 33 \pi \]
We can use \( \pi \approx 3.14 \) to get an approximate numerical value:
\[ V \approx 33 \times 3.14 \approx 103.26 , \text{cubic feet} \]
Volume of the Cylinder
You mentioned that the cylinder has a volume of 310.86 cubic feet. The formula for the volume of a cylinder is given by:
\[ V = \pi r^2 h \]
Where:
- The volume \( V \) is given as 310.86 cubic feet,
- The height \( h \) of the cylinder is 11 feet.
We can find the radius \( r \) of the cylinder using the volume formula. Rearranging the formula gives:
\[ r^2 = \frac{V}{\pi h} \]
Substituting in the values:
\[ r^2 = \frac{310.86}{\pi \cdot 11} \]
Using \( \pi \approx 3.14 \):
\[ r^2 = \frac{310.86}{3.14 \cdot 11} = \frac{310.86}{34.54} \approx 9.0 \]
Taking the square root gives:
\[ r \approx \sqrt{9.0} = 3 \text{ feet} \]
Summary
- The volume of the cone is approximately 103.26 cubic feet.
- The radius of the cylinder is approximately 3 feet, confirming that the radius of both the cone and cylinder is the same. The cylinder has a larger volume (310.86 cubic feet) due to its cylindrical shape and height.