Question

The radius (in cm) of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is :

• A. 4.2
• B. 2.1
• C. 8.4
• D. 1.5

Hint:

To fit the largest cone inside a cube: 1. The circular base of the cone will touch all four sides of one face of the cube. 2. This means the diameter of the cone's base is equal to the edge length of the cube. 3. Radius of cone = (Diameter of cone) / 2 = (Edge of cube) / 2. Given edge of cube = 4.2 cm. Radius of cone = \(4.2 / 2 = 2.1\) cm. The height of this largest cone would also be 4.2 cm.

Answer 1

The Correct Option is B

Concept: To cut the largest possible right circular cone from a cube, the base of the cone must be inscribed within one of the faces of the cube, and the height of the cone will be equal to the edge of the cube. Step 1: Visualize the situation Imagine a cube. The largest circular base of a cone that can fit on one face of the cube will have its diameter equal to the side length (edge) of that face. The height of this cone can be at most the edge of the cube if its apex touches the opposite face. Step 2: Relate the cone's dimensions to the cube's edge Let the edge of the cube be \(a\). Given, edge of the cube \(a = 4.2 \text{ cm}\). For the largest cone:
The diameter of the base of the cone will be equal to the edge of the cube, \(a\). Diameter of cone's base = \(a = 4.2 \text{ cm}\).
The radius (\(r_{cone}\)) of the base of the cone is half of its diameter. \(r_{cone} = \frac{\text{Diameter}}{2} = \frac{a}{2}\).
The height (\(h_{cone}\)) of the largest cone will be equal to the edge of the cube, \(a\). \(h_{cone} = a = 4.2 \text{ cm}\). (Though the height is not needed to find the radius of the base). Step 3: Calculate the radius of the cone Using \(r_{cone} = \frac{a}{2}\) and \(a = 4.2 \text{ cm}\): \[ r_{cone} = \frac{4.2 \text{ cm}}{2} \] \[ r_{cone} = 2.1 \text{ cm} \] The radius of the largest right circular cone that can be cut out is \(2.1 \text{ cm}\). This matches option (2).