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
In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.
What is the area (in cm²) of the rectangle PLMN?
Note: The figure shown is representative.
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius $r$ cm as shown in the figure. The side of the dodecagon is $d$ cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side $r$ cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.
- Which one of the following options has the correct sequence of objects arranged in the increasing number of mirror lines (lines of symmetry)?
- A circle with center at $(x, y) = (0.5, 0)$ and radius $= 0.5$ intersects with another circle with center at $(x, y) = (1, 1)$ and radius $= 1$ at two points. One of the points of intersection $(x, y)$ is:
- During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be $89.85^{\circ}$, the ratio of the Earth-Sun and Earth-Moon distances is closest to
