Answer :
To determine the volume of a cube given its surface area, we can follow these steps:
1. Understand the relationship between the surface area and the side length of a cube:
- The surface area \( S \) of a cube is given by \( 6a^2 \), where \( a \) is the length of the side of the cube.
2. Find the side length \( a \) of the cube:
- Given the surface area \( S = 294 \, \text{m}^2 \).
- Set up the equation: \( 6a^2 = 294 \).
- Solve for \( a^2 \):
[tex]\[ a^2 = \frac{294}{6} \][/tex]
- Simplify the calculation:
[tex]\[ a^2 = 49 \][/tex]
3. Determine the side length \( a \):
- Take the square root of both sides of the equation \( a^2 = 49 \):
[tex]\[ a = \sqrt{49} \][/tex]
- Therefore, \( a = 7 \, \text{m} \).
4. Calculate the volume \( V \) of the cube:
- The volume \( V \) of a cube is given by \( V = a^3 \).
- Using \( a = 7 \, \text{m} \):
[tex]\[ V = 7^3 \][/tex]
- Simplify the calculation:
[tex]\[ V = 343 \, \text{m}^3 \][/tex]
Thus, the volume of the cube is [tex]\( 343 \, \text{m}^3 \)[/tex].
1. Understand the relationship between the surface area and the side length of a cube:
- The surface area \( S \) of a cube is given by \( 6a^2 \), where \( a \) is the length of the side of the cube.
2. Find the side length \( a \) of the cube:
- Given the surface area \( S = 294 \, \text{m}^2 \).
- Set up the equation: \( 6a^2 = 294 \).
- Solve for \( a^2 \):
[tex]\[ a^2 = \frac{294}{6} \][/tex]
- Simplify the calculation:
[tex]\[ a^2 = 49 \][/tex]
3. Determine the side length \( a \):
- Take the square root of both sides of the equation \( a^2 = 49 \):
[tex]\[ a = \sqrt{49} \][/tex]
- Therefore, \( a = 7 \, \text{m} \).
4. Calculate the volume \( V \) of the cube:
- The volume \( V \) of a cube is given by \( V = a^3 \).
- Using \( a = 7 \, \text{m} \):
[tex]\[ V = 7^3 \][/tex]
- Simplify the calculation:
[tex]\[ V = 343 \, \text{m}^3 \][/tex]
Thus, the volume of the cube is [tex]\( 343 \, \text{m}^3 \)[/tex].
