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Chapter 10: Problem 49

Explain the following analogy: In terms of formulas used to compute volume, a pyramid is to a rectangular solid just as a cone is to a cylinder.

Short Answer

Expert verified
The analogy refers to the similarity in the formulas used to calculate volume. Just like a pyramid's volume is one-third of a rectangular solid's if they have the same base area and height (since we multiply by 1/3 for a pyramid), a cone's volume is one-third of a cylinder's given the same base area and height, which is why a pyramid is to a rectangular solid as a cone is to a cylinder in terms of volume formulas.

Step by step solution

01

Understand the Volume Formulas for Pyramid and Rectangular Solid

The volume, V, of a pyramid is given by \(V = \frac{1}{3} \times \text{base area} \times \text{height}\). On the other hand, for a rectangular solid (or any prism), it is given by \(V = \text{base area} \times \text{height}\).
02

Understand the Volume Formulas for Cone and Cylinder

The volume, V, of a cone is calculated as \(V = \frac{1}{3} \times \text{base area} \times \text{height}\). For a cylinder, the formula is \(V = \text{base area} \times \text{height}\).
03

Explain the Analogy

The analogy is showcasing the relationship between the volume formulas. The pyramid and the cone both have their volumes calculated as one-third the base area times the height, while the rectangular solid and the cylinder simply multiply the base area by the height. Therefore, in terms of volume formulas, a pyramid is to a rectangular solid as a cone is to a cylinder because the way we calculate their volumes are analogous.

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