Question

Find the volume of the largest right circular cone that can be cut out of cube having 5 cm as its length of the side.

• A. 21.82
• B. 45.67
• C. 32.72
• D. 65.45

Answer 1

Answer:

C. 32.72

Understanding the Geometry

• A right circular cone is a three-dimensional geometric shape with a circular base and a vertex that is directly above the center of the base.
• A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

Maximizing Cone Volume within a Cube

• To cut the largest possible right circular cone from a cube, the cone's base must be inscribed within one of the cube's faces, and the cone's height must be equal to the cube's side length.
• Diameter of the Cone's Base: The diameter of the largest circle that can be inscribed in a square face of the cube will be equal to the side length of the cube.
• Radius of the Cone's Base (r): If the side length of the cube is 'a', then the diameter of the cone's base is 'a'. Therefore, the radius (r) = a/2.
• Height of the Cone (h): The maximum height the cone can have within the cube is equal to the side length of the cube, so h = a.

Formula for the Volume of a Cone

• The volume (V) of a right circular cone is given by the formula: V = (1/3) * "π" * r^2 * h
• Where 'r' is the radius of the base and 'h' is the height of the cone.

Applying the Formula to the Given Problem

• Given side length of the cube (a) = 5 cm.
• Radius of the cone's base (r) = a/2 = 5/2 cm.
• Height of the cone (h) = a = 5 cm.
• Substitute these values into the volume formula:
• V = (1/3) * "π" * (5/2)^2 * 5
• V = (1/3) * "π" * (25/4) * 5
• V = (1/3) * "π" * (125/4)
• V = (125/12) * "π"

Calculating the Numerical Value

• Using the approximate value of "π" ≈ 3.14159
• V ≈ (125/12) * 3.14159
• V ≈ 10.4167 * 3.14159
• V ≈ 32.7249...
• Rounding to two decimal places, the volume is approximately 32.72 cubic centimeters.

Exam Tips

• Key Ratios: Always remember that for the largest cone in a cube, the radius is half the side length, and the height is equal to the side length.
• Formula Recall: Be quick to recall the volume of a cone formula.
• Approximation: For MCQs, often you can estimate "π" as 22/7 or 3.14 to quickly narrow down options if exact calculation isn't needed.
• Units: Ensure the final answer includes the correct cubic units (e.g., cm³).