To find the volume of the cone, we can use 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 of the cone.
Given that the diameter of the cone is 30 centimeters, we can find the radius:
\[ r = \frac{\text{diameter}}{2} = \frac{30}{2} = 15 \text{ cm} \]
The problem states that the height of the cone is twice the length of the diameter. Since the diameter is 30 cm:
\[ h = 2 \times \text{diameter} = 2 \times 30 = 60 \text{ cm} \]
Now, we can plug the values of \( r \) and \( h \) into the volume formula:
\[ V = \frac{1}{3} \pi (15)^2 (60) \]
Calculating \( r^2 \):
\[ 15^2 = 225 \]
Now substituting that into the equation for volume:
\[ V = \frac{1}{3} \pi (225) (60) \]
Calculating \( 225 \times 60 \):
\[ 225 \times 60 = 13500 \]
Now substitute this back into the volume equation:
\[ V = \frac{1}{3} \pi (13500) \]
Calculating \( \frac{13500}{3} \):
\[ \frac{13500}{3} = 4500 \]
So the volume is:
\[ V = 4500\pi \text{ cm}^3 \]
The answer is:
4,500π cm³