A sphere of radius \(r\) cm is packed in a box of cubical shape.
What should be the minimum volume (in cm\(^3\)) of the box that can enclose the sphere?
Show Hint
- \( \frac{r^3}{8} \)
- \( r^3 \)
- \( 2r^3 \)
- \( 8r^3 \)
The Correct Option is D
Solution and Explanation
Step 1: Understand the geometry of the problem.
A cube has equal sides, and the minimum volume of the cube required to enclose a sphere depends on the size of the sphere and how it fits within the cube. Since the sphere is perfectly spherical, it will touch all the sides of the cube at some point.
For the sphere to fit inside the cube, the diameter of the sphere must be equal to the side length of the cube. The diameter of the sphere is \(2r\), where \(r\) is the radius of the sphere.
Step 2: Determine the side length of the cube.
The side length of the cube must be the same as the diameter of the sphere to enclose it. Therefore, the side length of the cube is: \[ \text{side length of the cube} = 2r \]
Step 3: Calculate the volume of the cube.
The volume of a cube is given by the formula: \[ V = \text{side length}^3 \] Substituting the side length \(2r\) into the formula: \[ V = (2r)^3 = 8r^3 \] Thus, the minimum volume of the cube required to enclose the sphere is \(8r^3\).
Step 4: Analyze the options.
- (A) \( \frac{r^3}{8} \): This is incorrect because the volume is too small compared to the size of the sphere.
- (B) \( r^3 \): This is also incorrect. A volume of \(r^3\) would not be sufficient to enclose a sphere with radius \(r\).
- (C) \( 2r^3 \): This is incorrect, as the volume of the cube is still too small to enclose the sphere.
- (D) \( 8r^3 \): This is the correct option. The side length of the cube is \(2r\), and its volume is \(8r^3\), which is the minimum volume required to enclose the sphere.
Step 5: Conclusion.
The correct answer is (D) \(8r^3\). This is the minimum volume of the box that can enclose the sphere.
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