Question

explain how to get the formula for the volume of a cone using the relationship between cones and cylinders. use the drop - downs to complete the sentences. if you have a cylinder with a specific base area and height, it would take cones with the same base area and height have the same volume. the formula for the volume of a cylinder is ( v = bh ) or ( v=pi r^{2}h ). that means the formula for the volume of a cone is or .

Explanation:

Step1: Recall volume relationship

The volume of a cylinder with base area \(B\) (or \(\pi r^{2}\)) and height \(h\) is \(V_{cylinder}=Bh=\pi r^{2}h\). Through experiments (pouring sand or water from cones to a cylinder with the same base and height), it is found that the volume of a cone is one - third of the volume of a cylinder with the same base area and height.

Step2: Calculate number of cones

Let the volume of the cylinder be \(V_{cylinder}\) and the volume of the cone be \(V_{cone}\). If \(V_{cone}=\frac{1}{3}V_{cylinder}\), then \(V_{cylinder} = 3V_{cone}\). So, if you have a cylinder with a specific base area and height, it would take \(3\) cones with the same base area and height to have the same volume.

Step3: Derive cone volume formula

Since \(V_{cylinder}=Bh=\pi r^{2}h\) and \(V_{cone}=\frac{1}{3}V_{cylinder}\), the formula for the volume of a cone is \(V=\frac{1}{3}Bh\) or \(V = \frac{1}{3}\pi r^{2}h\)

Answer:

The first blank: \(3\); the second blank: \(\frac{1}{3}Bh\); the third blank: \(\frac{1}{3}\pi r^{2}h\)