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Describe an informal argument for the volume of a cone. If you know the volume of a cylinder, what must you do to find the volu...Question
Describe an informal argument for the volume of a cone. If you know the volume of a cylinder, what must you do to find the volume of a cone that has the same height and base radius?(1 point)ResponsesYou need to multiply the volume of the cylinder by 3.You need to multiply the volume of the cylinder by 3.You need to add 3 cubic units to the volume of the cylinder.You need to add 3 cubic units to the volume of the cylinder.You need to multiply the volume of the cylinder by 13.You need to multiply the volume of the cylinder by Start Fraction 1 over 3 End Fraction .You need to divide the volume of the cylinder by 13.
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GPT-4o mini
To find the volume of a cone that has the same height and base radius as a cylinder, we start by recalling the volume of a cylinder. The volume \( V \) of a cylinder is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
where \( r \) is the radius of the base and \( h \) is the height.
An informal argument for the volume of a cone can be made by thinking about how the cone fits within the cylinder. If you imagine filling the cylinder with water and then pouring out the water into a cone of the same radius and height, you would find that the cone can only hold a portion of the water that the cylinder can.
It is known that a cone with the same height and base radius as a cylinder has one-third the volume of that cylinder. This is because when you fill a cylinder with 3 cones of the same dimensions, they perfectly fill the cylinder, leaving no space. Hence, the volume of the cone can be determined by dividing the volume of the cylinder by 3.
As such, to find the volume of the cone, you need to multiply the volume of the cylinder by \( \frac{1}{3} \):
\[
V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}}
\]
Therefore, the correct response is:
**You need to multiply the volume of the cylinder by \( \frac{1}{3} \).**
\[
V_{\text{cylinder}} = \pi r^2 h
\]
where \( r \) is the radius of the base and \( h \) is the height.
An informal argument for the volume of a cone can be made by thinking about how the cone fits within the cylinder. If you imagine filling the cylinder with water and then pouring out the water into a cone of the same radius and height, you would find that the cone can only hold a portion of the water that the cylinder can.
It is known that a cone with the same height and base radius as a cylinder has one-third the volume of that cylinder. This is because when you fill a cylinder with 3 cones of the same dimensions, they perfectly fill the cylinder, leaving no space. Hence, the volume of the cone can be determined by dividing the volume of the cylinder by 3.
As such, to find the volume of the cone, you need to multiply the volume of the cylinder by \( \frac{1}{3} \):
\[
V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}}
\]
Therefore, the correct response is:
**You need to multiply the volume of the cylinder by \( \frac{1}{3} \).**