A sculptor is created a design by carving a cone out of a cylinder The cone and cylinder share the same radius and height if the volume of the cylinder before removing the cone is 54 in^3 what is the voluming of the amount remaining?

40 in^3
36 in^3
18 in^3
27 in^3

3 answers

To find the volume of the remaining material after carving out the cone from the cylinder, we first need to determine the volume of the cone.

  1. Volume of the Cylinder: The volume of the cylinder is already given as 54 in³.

  2. Volume of the Cone: The formula for the volume of a cone is given by: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

    Since the cone and the cylinder have the same radius \( r \) and height \( h \), we need to express the volume of the cone in terms of the volume of the cylinder.

    The formula for the volume of the cylinder is: \[ V_{\text{cylinder}} = \pi r^2 h \]

  3. Relating the Volumes: From the given volume of the cylinder: \[ 54 = \pi r^2 h \]

    Therefore, \( \pi r^2 h = 54 \).

    Now substituting into the cone volume formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} (54) = 18 \text{ in}^3 \]

  4. Volume of the Remaining Material: The volume of the remaining material after carving out the cone is: \[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 54 \text{ in}^3 - 18 \text{ in}^3 = 36 \text{ in}^3 \]

Thus, the volume of the amount remaining is 36 in³.

gotta make sure are you correct-?

Let's double-check the calculations step-by-step to ensure accuracy.

  1. Volume of the Cylinder: The volume of the cylinder is given as \( V_{\text{cylinder}} = 54 , \text{in}^3 \).

  2. Volume of the Cone: The formula for the volume of a cone is: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h. \] The volume of the cylinder is expressed as: \[ V_{\text{cylinder}} = \pi r^2 h = 54 , \text{in}^3. \] Therefore, we can substitute this into the cone's formula: \[ V_{\text{cone}} = \frac{1}{3} \times 54 , \text{in}^3 = 18 , \text{in}^3. \]

  3. Calculating the Volume of the Remaining Material: The volume of the remaining material after removing the cone from the cylinder is: \[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 54 , \text{in}^3 - 18 , \text{in}^3 = 36 , \text{in}^3. \]

So, after reviewing the calculations, the concluding answer is indeed correct: the volume of the amount remaining after carving out the cone is 36 in³.