a sculptor is creating a design by carvinv a cone out of a cylinder. The cone and cylinder share the same radius and height. if the volume remaining is 144 in. ^3, what was the volume of the origional cylinnder?

1 answer

To solve this problem, we need to use the formulas for the volume of a cylinder and a cone.

  1. Volume of a Cylinder (\( V_c \)): \[ V_c = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.

  2. Volume of a Cone (\( V_{cone} \)): \[ V_{cone} = \frac{1}{3} \pi r^2 h \]

Given that the cone is carved out of the cylinder and they share the same radius \( r \) and height \( h \), we can express the remaining volume after carving the cone out of the cylinder as follows:

\[ \text{Volume remaining} = V_c - V_{cone} \]

Substituting in the formulas for the volumes:

\[ \text{Volume remaining} = \pi r^2 h - \frac{1}{3} \pi r^2 h \]

Factoring out \( \pi r^2 h \):

\[ \text{Volume remaining} = \pi r^2 h \left(1 - \frac{1}{3}\right) = \pi r^2 h \left(\frac{2}{3}\right) \]

Given that the volume remaining is \( 144 , \text{in}^3 \):

\[ \frac{2}{3} \pi r^2 h = 144 \]

To find the original volume of the cylinder:

\[ V_c = \pi r^2 h = \frac{3}{2} \times 144 = 216 , \text{in}^3 \]

Thus, the volume of the original cylinder is:

\[ \boxed{216 , \text{in}^3} \]