The volume of a cube is 2.197 cm3. Find the side of the cube.
1.3 cm
Finding the Side Length of a Cube from its Volume
The question asks us to find the side length of a cube given its volume. We are provided with the volume of the cube as 2.197 cm<sup>3</sup>.
Understanding Cube Volume
A cube is a three-dimensional solid figure with six equal square faces. All edges (sides) of a cube are equal in length.
The formula for the volume (V) of a cube with side length (s) is:
\( V = s^3 \)
In this problem, we are given the volume \( V \) and need to find the side length \( s \). To do this, we need to find the cube root of the volume.
\( s = \sqrt[3]{V} \)
Step-by-Step Calculation of the Side
Given volume \( V = 2.197 \text{ cm}^3 \).
We need to find the value of \( s \) such that \( s^3 = 2.197 \).
So, \( s = \sqrt[3]{2.197} \)
Let's consider the options provided:
- 1.3 cm
- 1.4 cm
- 1.1 cm
- 1.2 cm
We can test each option by cubing it (multiplying it by itself three times) to see which one gives 2.197.
- If \( s = 1.1 \text{ cm} \), \( s^3 = 1.1 \times 1.1 \times 1.1 = 1.21 \times 1.1 = 1.331 \text{ cm}^3 \)
- If \( s = 1.2 \text{ cm} \), \( s^3 = 1.2 \times 1.2 \times 1.2 = 1.44 \times 1.2 = 1.728 \text{ cm}^3 \)
- If \( s = 1.3 \text{ cm} \), \( s^3 = 1.3 \times 1.3 \times 1.3 = 1.69 \times 1.3 = 2.197 \text{ cm}^3 \)
- If \( s = 1.4 \text{ cm} \), \( s^3 = 1.4 \times 1.4 \times 1.4 = 1.96 \times 1.4 = 2.744 \text{ cm}^3 \)
From the calculations, we see that when the side length \( s \) is 1.3 cm, the volume \( s^3 \) is 2.197 cm<sup>3</sup>.
Therefore, the side of the cube is 1.3 cm.
Verification
Volume = \( (\text{Side})^3 \)
Volume = \( (1.3 \text{ cm})^3 \)
Volume = \( 1.3 \text{ cm} \times 1.3 \text{ cm} \times 1.3 \text{ cm} \)
Volume = \( 1.69 \text{ cm}^2 \times 1.3 \text{ cm} \)
Volume = \( 2.197 \text{ cm}^3 \)
This matches the given volume.
The side of the cube is 1.3 cm.
| Given | Formula Used | Calculation | Result |
|---|---|---|---|
| Volume = 2.197 cm<sup>3</sup> | \( V = s^3 \), \( s = \sqrt[3]{V} \) | \( s = \sqrt[3]{2.197} \) | \( s = 1.3 \) cm |
Revision Table: Cube Properties
| Property | Formula (Side = \(s\)) | Description |
|---|---|---|
| Volume | \( V = s^3 \) | Space occupied by the cube. |
| Surface Area (Total) | \( A = 6s^2 \) | Sum of the areas of all 6 square faces. |
| Surface Area (Lateral) | \( A_{lateral} = 4s^2 \) | Sum of the areas of the 4 side faces. |
| Diagonal of a face | \( d_{face} = s\sqrt{2} \) | Diagonal across one of the square faces. |
| Diagonal of the cube | \( d_{cube} = s\sqrt{3} \) | Diagonal connecting opposite vertices through the interior. |
Additional Information: Cube Roots and Perfect Cubes
Finding the side of a cube from its volume involves calculating the cube root. The cube root of a number is the value that, when multiplied by itself three times, gives the original number.
For example:
- \( \sqrt[3]{8} = 2 \) because \( 2 \times 2 \times 2 = 8 \)
- \( \sqrt[3]{27} = 3 \) because \( 3 \times 3 \times 3 = 27 \)
Numbers like 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, etc., are called perfect cubes because their cube roots are integers.
In this problem, 2.197 is the cube of 1.3. We can think of 2197 (without the decimal) and recognize it as a perfect cube: \( 13 \times 13 \times 13 = 2197 \). Since 2.197 has three decimal places, its cube root will have one decimal place, which is 1.3.
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:
