Answer :
To solve this problem, we'll go through each step methodically:
1. Understand the surface area of a cube:
The surface area \( A \) of a cube is given by the formula:
[tex]\[ A = 6a^2 \][/tex]
where \( a \) is the length of a side of the cube.
2. Given information:
The surface area of the cube is \( 294 \, \text{m}^2 \).
3. Set up the equation:
We substitute the given surface area into the surface area formula:
[tex]\[ 6a^2 = 294 \][/tex]
4. Solve for \( a^2 \):
Divide both sides by 6 to isolate \( a^2 \):
[tex]\[ a^2 = \frac{294}{6} \][/tex]
[tex]\[ a^2 = 49 \][/tex]
5. Solve for \( a \):
Take the square root of both sides to find \( a \):
[tex]\[ a = \sqrt{49} \][/tex]
[tex]\[ a = 7 \, \text{m} \][/tex]
6. Find the volume \( V \):
The volume of a cube is given by the formula:
[tex]\[ V = a^3 \][/tex]
Substitute \( a = 7 \, \text{m} \) into the volume formula:
[tex]\[ V = 7^3 \][/tex]
[tex]\[ V = 343 \, \text{m}^3 \][/tex]
Therefore, the volume of the cube is [tex]\( 343 \, \text{m}^3 \)[/tex].
1. Understand the surface area of a cube:
The surface area \( A \) of a cube is given by the formula:
[tex]\[ A = 6a^2 \][/tex]
where \( a \) is the length of a side of the cube.
2. Given information:
The surface area of the cube is \( 294 \, \text{m}^2 \).
3. Set up the equation:
We substitute the given surface area into the surface area formula:
[tex]\[ 6a^2 = 294 \][/tex]
4. Solve for \( a^2 \):
Divide both sides by 6 to isolate \( a^2 \):
[tex]\[ a^2 = \frac{294}{6} \][/tex]
[tex]\[ a^2 = 49 \][/tex]
5. Solve for \( a \):
Take the square root of both sides to find \( a \):
[tex]\[ a = \sqrt{49} \][/tex]
[tex]\[ a = 7 \, \text{m} \][/tex]
6. Find the volume \( V \):
The volume of a cube is given by the formula:
[tex]\[ V = a^3 \][/tex]
Substitute \( a = 7 \, \text{m} \) into the volume formula:
[tex]\[ V = 7^3 \][/tex]
[tex]\[ V = 343 \, \text{m}^3 \][/tex]
Therefore, the volume of the cube is [tex]\( 343 \, \text{m}^3 \)[/tex].
