Question

a baker is designing a cone - shaped funnel to pour batter into cylinder - shaped molds. the funnel and molds have the same base area and height. to determine how much batter the funnel can hold, the baker needs to compare the volume of the cone to the volume of the cylinder. explain how to derive the formula for the volume of a cone using the relationship between the volume of a cone and a cylinder. drag and drop the values to complete the sentences. if a cylinder and a cone have the same base area and height, it would take cones to fill the volume of the cylinder. the formula for the volume of a cylinder is ( v = bh ) or. so, the volume of a cone can be expressed as

Explanation:

Step1: Recall volume formulas

The volume of a cylinder \(V_{cylinder}=Bh\) where \(B\) is the base area and \(h\) is the height. For a circle - shaped base, \(B = \pi r^{2}\), so \(V_{cylinder}=\pi r^{2}h\).

Step2: Use the relationship between cone and cylinder

Through experiments (e.g., filling a cone with sand and pouring it into a cylinder of the same base and height), it is found that it takes 3 cones to fill the cylinder. So \(V_{cone}=\frac{1}{3}V_{cylinder}\).

Step3: Derive the cone volume formula

Since \(V_{cylinder}=Bh\) (or \(\pi r^{2}h\)), then \(V_{cone}=\frac{1}{3}Bh\) (or \(\frac{1}{3}\pi r^{2}h\))

Answer:

It would take 3 cones to fill the volume of the cylinder. The formula for the volume of a cylinder is \(V = Bh\) or \(\pi r^{2}h\). So, the volume of a cone can be expressed as \(\frac{1}{3}Bh\) (or \(\frac{1}{3}\pi r^{2}h\))