Question

What is the length of the edge of the largest cube that can be placed inside a sphere of radius 10 cm?

• A. \( 10 / (\sqrt{2}) \) cm
• B. \( 10 / (\sqrt{3}) \) cm
• C. \( 20 / (\sqrt{2}) \) cm
• D. \( 20 / (\sqrt{3}) \) cm

Hint:

For problems involving a 3D shape inscribed in another (e.g., cube in a sphere, sphere in a cube, cone in a sphere), always identify the longest line segment in the inner shape that connects two points on the outer shape. This relationship typically provides the equation needed to solve the problem.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the side length of the largest possible cube that can be inscribed within a sphere. For the cube to be the largest, its vertices must touch the inner surface of the sphere.
Step 2: Key Formula or Approach:
The key insight is that the main diagonal (or space diagonal) of the inscribed cube is equal to the diameter of the sphere.
Let 'a' be the length of the edge of the cube.
The formula for the main diagonal of a cube is \(d = a\sqrt{3}\).
Let 'R' be the radius of the sphere. The diameter of the sphere is \(2R\).
Therefore, the governing equation is \(a\sqrt{3} = 2R\).
Step 3: Detailed Explanation:
We are given the radius of the sphere, \(R = 10\) cm.
First, calculate the diameter of the sphere:
\[ \text{Diameter} = 2R = 2 \times 10 = 20 \text{ cm} \] Now, we equate the main diagonal of the cube to the diameter of the sphere:
\[ a\sqrt{3} = 20 \] To find the length of the edge 'a', we solve for 'a':
\[ a = \frac{20}{\sqrt{3}} \text{ cm} \] Step 4: Final Answer:
The length of the edge of the largest cube that can be placed inside the sphere is \( \frac{20}{\sqrt{3}} \) cm. This corresponds to option (D).