Question: A cube with an edge length of 6 contains the largest possible sphere that completely fits inside of it. If the volume V of a sphere with radius r is given by the formula
V = \(\frac{4}{3}\pi r^3\), what is the volume of the empty space inside the cube?

1. \(36(\pi^3 - 6)\)
2. \(36\pi\)
3. \(36=(6-\pi)\)
4. \(36(\pi-1)\)
5. \(36(4-\pi)\)

Correct Answer: C
Solution and Explanation:

Approach Solution 1:

The largest possible sphere that fits inside the cube has its diameter equal to the length of the edge of the cube. Since the edge of the cube has a length of 6, the diameter of the sphere also has a length of 6, and thus its radius has a length of 3.

Since the volume of the cube is \(6^3=216\) and that of the sphere is \(\frac{4}{3}*\pi*3^3=36\pi\)

The volume of the empty space inside the cube is \(216 - 36\pi= 36(6 -\pi)\)

Approach Solution 2:

Since the largest sphere is fitted inside the cube of 6 cm; then its diameter = 6 cm or radius = 3 cm

Area of the sphere = \(\frac{4}{3}*27*\pi=36\pi\)

And the area of cube = \(6^3 = 216\)

So the empty space = \(216-36\pi\)

Required answer = \(36(6 - \pi)\)

Approach Solution 3:

The problem statement states that:

Given:

• A cube with an edge length of 6 contains the largest possible sphere that completely fits inside of it.
• The volume V of a sphere with radius r is given by the formula V = \(\frac{4}{3}\pi r^3\)

Find Out:

• The volume of the empty space inside the cube.

Let the largest possible sphere of radius "r" completely fits inside of the cube of side "a".
Therefore, we can say as per the mathematical formula:
2r =a =6
=> r=3.

Empty space inside cube = cube volume - sphere volume
Therefore, Empty space inside cube = \(a^3− \frac{4}{3}π r^3= 6^3 − \frac{4}{3}π 3^3 = 6^3 − 6^2π = 36(6−π)\)

Hence, Empty space inside cube = \(36(6−π)\)

“A cube with an edge length of 6 contains the largest possible sphere”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The candidates can go through GMAT Quant practice papers to strenthen their mathematical understanding.

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