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
The area of the floor of a cubical room is 192 m 2. The length of the longest rod that can be kept in that room is :
A solid metallic rectangular block of dimensions 112 cm × 44 cm × 25 cm is melted and recast into a cylinder of radius 35 cm. The curved surface area (in cm 2) of the cylinder is: (Take π = 22/7)
If the volume of a cube is 175616 cm 3, what is its side?
The volume of a right circular cone is 1232 cm 3. If the height of the cone is 24 cm, then what will be the radius of its base?
A right triangle contains the right angle between the sides 5 cm and 7 cm. A cone is generated by revolving about the side 5 cm. The volume of this cone is:
