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?
Minimum Volume of Cubical Box Enclosing Sphere
To find the minimum volume of a cubical box that can enclose a sphere, we need to determine the smallest possible dimensions of the cube.
Relating Sphere Diameter to Cube Side
- The sphere has a radius of $r$ cm.
- The diameter of the sphere is twice the radius, which is $2r$ cm.
- For a cubical box to completely enclose the sphere, the length of each side of the cube must be at least equal to the diameter of the sphere.
- Therefore, the minimum side length ($s$) of the cubical box is $s = 2r$ cm.
Calculating Minimum Box Volume
The volume of a cube is calculated as side length cubed ($V = s^3$).
- Using the minimum side length $s = 2r$, the minimum volume ($V_{min}$) of the cubical box is: $ V_{min} = s^3 = (2r)^3 $
- Calculating the cube: $ V_{min} = 2^3 \times r^3 = 8r^3 $
Thus, the minimum volume of the cubical box required is $8r^3$ cubic centimeters.
Answer Explanation
The minimum volume required for the cubical box is achieved when its side length is exactly equal to the diameter of the sphere ($2r$). The volume calculation $(2r)^3$ correctly yields $8r^3$. Other options represent volumes that are too small to enclose the sphere.
Important Questions from Mensuration and Geometry
The city of Atlantis was crafted by the God of the seas, Poseidon. It was made of alternating concentric circular rings of land (shaded) and water (not shaded) as represented in the figure (not to scale). The radius of Inner Island was 2.5 stades (a unit of length used in ancient Greece). The water surrounding Inner Island was one stade wide (length AB). This was surrounded by two pairs of alternating rings of land and water. The first pair of land and water was two stades wide each (lengths BC and CD), and the outer pair is three stades wide each (lengths DE and EF).
The ratio of the surface area of the land to that of the water in the city of Atlantis is _________ (round off to two decimal places).
In the given figure, $P, Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45^\circ$. The figure is representative.
The area of the shaded region $PQRO$ is ______________ cm$^2$.
- A straight line $y = x - 1$ intersects a circle with center at $x = 1, y = 1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is _______. (rounded off to three decimal places)
- The shell of a hollow spherical nanoparticle has a uniform thickness of 3 nanometers (nm). The outer radius of the nanoparticle is 5 nm. The ratio of the volume of the shell to the volume of the hollow core is ________
(Round off to one decimal place) - The volume of a sphere of diameter 1 unit is ______ than the volume of a cube of side 1 unit.
