A hollow cube is made of paper to have a volume of 512 cubic units. How much paper in square units will be required to make the cube?
384
Calculating Paper Needed for a Hollow Cube
The problem asks us to find the amount of paper required to make a hollow cube with a given volume. The amount of paper needed corresponds to the total surface area of the cube.
Understanding Cube Properties
A cube is a three-dimensional shape with six equal square faces. If 's' is the length of one side (or edge) of the cube:
- The volume (V) of a cube is given by the formula: \(V = s^3\)
- The surface area (A) of a cube (total area of all six faces) is given by the formula: \(A = 6s^2\)
Step-by-Step Solution
We are given that the volume of the hollow cube is 512 cubic units. We can use the volume formula to find the length of the side (s) of the cube.
Step 1: Find the side length (s) from the volume
Given Volume, \(V = 512\) cubic units.
Using the volume formula: \(V = s^3\)
So, \(s^3 = 512\)
To find 's', we need to calculate the cube root of 512:
\(s = \sqrt[3]{512}\)
We need to find a number that, when multiplied by itself three times, equals 512.
\(8 \times 8 \times 8 = 64 \times 8 = 512\)
Therefore, the side length of the cube is \(s = 8\) units.
Step 2: Calculate the surface area using the side length
Now that we have the side length, \(s = 8\) units, we can calculate the surface area (A) of the cube using the surface area formula:
\(A = 6s^2\)
Substitute the value of 's' into the formula:
\(A = 6 \times (8)^2\)
\(A = 6 \times 64\)
Calculate the product:
\(A = 384\)
The surface area of the cube is 384 square units. This is the amount of paper required to make the hollow cube.
Final Answer
The amount of paper required to make the hollow cube is 384 square units.
| Property | Formula | Calculated Value |
|---|---|---|
| Volume (V) | \(s^3\) | 512 cubic units (Given) |
| Side Length (s) | \(\sqrt[3]{V}\) | 8 units |
| Surface Area (A) | \(6s^2\) | 384 square units |
Revision Table: Cube Formulas
| Cube Property | Formula |
|---|---|
| Side Length | s |
| Volume | \(V = s^3\) |
| Surface Area | \(A = 6s^2\) |
| Diagonal of a face | \(d_{face} = s\sqrt{2}\) |
| Space Diagonal | \(d_{space} = s\sqrt{3}\) |
Additional Information: Understanding 3D Shapes
This problem involves basic concepts of geometry, specifically dealing with three-dimensional shapes like a cube. Understanding how volume and surface area relate to the dimensions of a shape is crucial. Volume measures the space inside a 3D object, while surface area measures the total area of all the faces on its surface. For a hollow object made of material like paper, the amount of material needed usually refers to its surface area.
Cubes are a type of regular hexahedron, one of the five Platonic solids. They are fundamental shapes in geometry and are used in many practical applications and theoretical concepts.
Important Questions from Solid Figures
A cylindrical tube, open at both ends, is made of a metal sheet which is 0.5 cm thick. Its outer radius is 4 cm and length is 2 m. How much metal (in cm 3) has been used in making the tube?
The volume of a right circular cone is 308 cm 3 and the radius of its base is 7 cm. What is the curved surface area (in cm 2) of the cone? (Take π = \(\frac{22}{7} \) )
The slant height and radius of a right circular cone are in the ratio 29 ∶ 20. If its volume is 4838.4 π cm 3, then its radius is:
Six cubes, each of edge 2 cm, are joined end to end. What is the total surface area of the resulting cuboid in cm 2?
A solid cube of side 8 cm is dropped into a rectangular container of length 16 cm, breadth 8 cm and height 15 cm which is partly filled with water. If the cube is completely submerged, then the rise of water level (in cm) is:
