A given right circular cone has a volume p, and the largest...

A given right circular cone has a volume $p$, and the largest right circular cylinder that can be inscribed in the cone has a volume $\mathrm{q}$. Then $\mathrm{p}: \mathrm{q}$ is
(A) $9 ; 4$
(B) $8: 3$
(C) $7: 2$
(D) None of these

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Key Concepts

Volume Formulas for Solids

Understanding the volume formulas for solids such as cones and cylinders is essential. The volume of a cone is computed as one-third of the product of its base area and its height, while the volume of a cylinder is the product of its base area and its height. These formulas serve as the foundational tools for comparing the volumes of different three-dimensional shapes.

Geometric Similarity

When a cylinder is inscribed in a cone, similar triangles help relate the dimensions of the cylinder to those of the cone. This concept of geometric similarity allows the ratios of corresponding dimensions (such as radius and height) to be established, which is crucial for expressing the cylinder's volume in terms of the cone's dimensions.

Optimization Using Calculus

Determining the largest possible volume of an inscribed cylinder requires an optimization approach. By expressing the cylinder's volume as a function of a variable that respects the geometric constraints imposed by the cone (often through the similarity of triangles), calculus techniques such as differentiation can be applied to find the maximum volume. This optimization process is central to solving problems of this type.

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