A cube is placed inside a right circular cone of radius 20 cm and height 10 cm. One face of the cube rests on the base of the cone and all four vertices of the opposite face touch the curved surface of the cone. What is the length, in centimetres, of the side of the cube?

Introduction / Context:
This is a three dimensional geometry problem involving a cube inscribed inside a right circular cone. One face of the cube lies on the circular base of the cone and the opposite face touches the curved surface. The question tests understanding of similar triangles inside cones, the linear variation of radius with height in a cone, and the relationship between the geometry of the cube and the geometry of the cone.

Given Data / Assumptions:

• Right circular cone with base radius R = 20 cm.
• Height of the cone H = 10 cm.
• A cube of side length s is placed so that one face lies on the base of the cone.
• The four vertices of the opposite face of the cube lie exactly on the curved surface of the cone.
• The cube is oriented with its edges parallel to the cone axis and to the base diameter directions.

Concept / Approach:
The radius of the cone decreases linearly from R at the base to 0 at the vertex. At a height h above the base, the radius of the cone cross section is given by r(h) = R * (H - h) / H when the vertex is at the top. The top face of the cube is at height s above the base, and each vertex of this top face is at a distance s / sqrt(2) from the axis of the cone. For those vertices to lie on the cone surface, that radial distance must match the cone radius at that height. This condition gives an equation in s, which we solve and then approximate numerically.

Step-by-Step Solution:
Step 1: At the base, radius of the cone is R = 20 cm and height is H = 10 cm.Step 2: At a height h above the base, the cone radius is r(h) = R * (H - h) / H = 20 * (10 - h) / 10 = 2 * (10 - h).Step 3: The cube has side s, so the top face is at height h = s above the base.Step 4: The vertices of the top face lie on a circle of radius s / sqrt(2) from the axis, since a square of side s has diagonal s * sqrt(2) and half of that gives the distance from center to a vertex.Step 5: For each vertex to lie on the cone surface, we must have s / sqrt(2) = 2 * (10 - s).Step 6: Solve this equation: s / sqrt(2) = 20 - 2s, so s = (20 - 2s) * sqrt(2). Rearranging, s + 2 * sqrt(2) * s = 20 * sqrt(2), hence s * (1 + 2 * sqrt(2)) = 20 * sqrt(2).Step 7: Therefore s = (20 * sqrt(2)) / (1 + 2 * sqrt(2)). Numerically this is approximately 7.39 cm, which we round to 7.4 cm.

Verification / Alternative check:
We can quickly verify by substituting s approximately equal to 7.4 back into the condition s / sqrt(2) and comparing it with 2 * (10 - s).Compute s / sqrt(2) with s around 7.4 and also compute 2 * (10 - 7.4). These values are very close, which confirms that the vertices of the upper face lie nearly on the cone surface.Because the relationship between height and radius in a cone is linear, small rounding changes do not significantly affect the correctness of the model, and 7.4 cm is an appropriate rounded answer.

Why Other Options Are Wrong:
Values such as 5 cm or 6 cm give a cube that is too small; its top vertices would lie strictly inside the cone and would not touch the surface.Values like 8 cm or 9 cm give a cube that is too tall; the required cross sectional radius at the cube top becomes too small, and the vertices would lie outside the cone surface.Only the option 7.4 cm satisfies the geometric condition derived from similar triangles in the cone.

Common Pitfalls:
A typical mistake is to assume that the side of the cube is proportional directly to the cone radius without accounting for the actual height at which the top face lies.Another error is to use the base radius R = 20 cm directly as the radial distance for the top vertices, ignoring the linear radius decrease with height.Learners may also confuse the diagonal of the square face with its half, forgetting that the vertex distance from the center is s * sqrt(2) / 2, not s * sqrt(2).

Final Answer:
The length of the side of the cube is approximately 7.4 cm.