To solve the problem step by step, we will calculate the volume of the cube and its surface area based on the information provided.
### Step 1: Calculate the volume of water in the container before the cube is submerged.
The base of the container is a square with a side of 15 cm, and the water level is 3 cm from the top. Since the container is rectangular with a square base, we can calculate the volume of the water that fills the height of 3 cm.
**Volume of water (V_water) = Area of base × Height**
- Area of base = side × side = 15 cm × 15 cm = 225 cm²
- Height of water = 3 cm
So,
\[ V_{water} = 225 \, \text{cm}^2 \times 3 \, \text{cm} = 675 \, \text{cm}^3 \]
### Step 2: Calculate the total volume of water after the cube is submerged.
According to the problem, when the cube is submerged, 54 cm³ of water overflows. Therefore, the total volume of water after the cube is submerged can be calculated as follows:
**Total Volume after submerging the cube = Volume of water + Volume of overflow**
\[ V_{total} = V_{water} + V_{overflow} \]
\[ V_{total} = 675 \, \text{cm}^3 + 54 \, \text{cm}^3 = 729 \, \text{cm}^3 \]
### Step 3: Relate the total volume to the volume of the cube.
The volume of the cube (V_cube) is equal to the total volume of water after the cube is submerged. Therefore, we have:
\[ V_{cube} = V_{total} = 729 \, \text{cm}^3 \]
### Step 4: Calculate the side length of the cube.
Since the volume of a cube is given by the formula:
\[ V_{cube} = x^3 \]
where \( x \) is the side length of the cube, we can set up the equation:
\[ x^3 = 729 \]
To find \( x \), we take the cube root of both sides:
\[ x = \sqrt[3]{729} = 9 \, \text{cm} \]
### Step 5: Calculate the surface area of the cube.
The surface area (SA) of a cube is given by the formula:
\[ SA = 6 \times x^2 \]
Substituting the value of \( x \):
\[ SA = 6 \times (9 \, \text{cm})^2 = 6 \times 81 \, \text{cm}^2 = 486 \, \text{cm}^2 \]
### Final Results:
- Volume of the cube = 729 cm³
- Surface area of the cube = 486 cm²
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