Question
A sculptor is creating a design by carving 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 original cylinder?(1 point)
Responses
180 in.3
180 in. cubed
192 in.3
192 in. cubed
288 in.3
288 in. cubed
216 in.3
, what was the volume of the original cylinder?(1 point)
Responses
180 in.3
180 in. cubed
192 in.3
192 in. cubed
288 in.3
288 in. cubed
216 in.3
Answers
Answered by
GPT-4o mini
To find the volume of the original cylinder, we first need to calculate the volume of the cone that is carved out of it.
The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Given that the cone and cylinder have the same radius \( r \) and height \( h \), we can express the volume of the cylinder in terms of the volume of the cone.
If the volume remaining after carving the cone out of the cylinder is 144 in³, we can set up an equation:
\[ V_{\text{cylinder}} - V_{\text{cone}} = 144 \]
Substituting the formulas for the volumes, we get:
\[ \pi r^2 h - \frac{1}{3} \pi r^2 h = 144 \]
Factoring out \( \pi r^2 h \):
\[ \left(1 - \frac{1}{3}\right) \pi r^2 h = 144 \]
\[ \frac{2}{3} \pi r^2 h = 144 \]
To find the volume of the cylinder, we multiply both sides of the equation by \( \frac{3}{2} \):
\[ \pi r^2 h = 144 \times \frac{3}{2} = 216 \]
Thus, the volume of the original cylinder is:
**216 in³**
So the correct response is:
**216 in.3**
The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Given that the cone and cylinder have the same radius \( r \) and height \( h \), we can express the volume of the cylinder in terms of the volume of the cone.
If the volume remaining after carving the cone out of the cylinder is 144 in³, we can set up an equation:
\[ V_{\text{cylinder}} - V_{\text{cone}} = 144 \]
Substituting the formulas for the volumes, we get:
\[ \pi r^2 h - \frac{1}{3} \pi r^2 h = 144 \]
Factoring out \( \pi r^2 h \):
\[ \left(1 - \frac{1}{3}\right) \pi r^2 h = 144 \]
\[ \frac{2}{3} \pi r^2 h = 144 \]
To find the volume of the cylinder, we multiply both sides of the equation by \( \frac{3}{2} \):
\[ \pi r^2 h = 144 \times \frac{3}{2} = 216 \]
Thus, the volume of the original cylinder is:
**216 in³**
So the correct response is:
**216 in.3**
There are no human answers yet. A form for humans to post answers is coming very soon!