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)
Responses

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 3.
You need to multiply the volume of the cylinder by 3.

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 1/3
.

Answer 1

Correct answer: You need to multiply the volume of the cylinder by 1/3.

Informal argument(s):
- Pouring argument: If you fill a cone with water and pour into a cylinder with the same base and height, three identical cones exactly fill the cylinder. So the cone's volume is one third of the cylinder's volume.
- Slicing/Cavalieri argument: Slice both solids into very thin horizontal disks. A disk in the cylinder has area πr^2 (constant). A disk in the cone at height z has radius proportional to (1 − z/h), so its area is πr^2(1 − z/h)^2. Integrating these disk areas from 0 to h gives the cone's volume = (1/3)πr^2h.

Thus V_cone = (1/3) V_cylinder = (1/3)πr^2h.