A rectangular container, whose base is a square of side 15cm, stands on a horizontal table and holds water upto 3cm from the top. When a cube is placed in the water and is completely submerged, the water rises to the top and `54cm^(3)` of water overflows. Calculate the volume of the cube and its surface area.

Answer

To solve the problem step by step, we will follow the given information and apply the necessary formulas.

### Step 1: Calculate the volume of the rectangular container.
The base of the container is a square with a side length of 15 cm, and the height of the water is 3 cm from the bottom.

**Volume of the container (V) = Base Area × Height**
- Base Area = Side × Side = 15 cm × 15 cm = 225 cm²
- Height of the water = 3 cm

So,
\[ V = 225 \, \text{cm}^2 \times 3 \, \text{cm} = 675 \, \text{cm}^3 \]

### Step 2: Understand the effect of the submerged cube.
When the cube is submerged, it causes 54 cm³ of water to overflow. This means the volume of the cube is equal to the volume of water that overflows plus the volume of water already in the container.

Let the volume of the cube be \( V_{cube} \).
\[ V_{cube} = V_{container} + \text{Overflow} \]
\[ V_{cube} = 675 \, \text{cm}^3 + 54 \, \text{cm}^3 = 729 \, \text{cm}^3 \]

### Step 3: Calculate the side length of the cube.
The volume of a cube is given by the formula:
\[ V_{cube} = a^3 \]
where \( a \) is the side length of the cube.

We have:
\[ a^3 = 729 \]

To find \( a \), we take the cube root of 729:
\[ a = \sqrt[3]{729} = 9 \, \text{cm} \]

### Step 4: Calculate the surface area of the cube.
The surface area (SA) of a cube is given by the formula:
\[ SA = 6a^2 \]
Substituting \( a = 9 \, \text{cm} \):
\[ SA = 6 \times (9 \, \text{cm})^2 = 6 \times 81 \, \text{cm}^2 = 486 \, \text{cm}^2 \]

### Final Results:
- Volume of the cube = 729 cm³
- Surface area of the cube = 486 cm²

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