A rectangular fish tank fish has a base 2 feet wide and 2 feet long. When the tank is partially filled with water, a solid cube with an edge length of 1 foot is placed in the tank. If no overflow of water from the tank is assumed, by how many inches will be level of the water in the tank rise when the cube becomes completely submerged?

Options

A. `(1)/(6)`
B. `(1)/(2)`
C. `2`
D. `3`

Answer

To solve the problem of how much the water level in the tank rises when a solid cube with an edge length of 1 foot is submerged, we can follow these steps:

### Step 1: Calculate the Volume of the Cube
The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]
Given that the edge length of the cube is 1 foot, we convert this to inches (1 foot = 12 inches):
\[ V = 12 \times 12 \times 12 = 1728 \text{ cubic inches} \]

**Hint:** Remember to convert all measurements to the same unit before performing calculations.

### Step 2: Determine the Base Area of the Tank
The base area \( A \) of the rectangular tank can be calculated using the formula:
\[ A = \text{width} \times \text{length} \]
The width and length of the tank are both 2 feet, which is equivalent to 24 inches. Therefore:
\[ A = 24 \times 24 = 576 \text{ square inches} \]

**Hint:** Ensure you square the dimensions when calculating area.

### Step 3: Relate the Volume of the Displaced Water to the Rise in Water Level
When the cube is submerged, it displaces an amount of water equal to its volume. The volume of water displaced will cause the water level to rise. The volume of the displaced water can also be expressed in terms of the base area and the height of the rise \( h \):
\[ \text{Volume of displaced water} = A \times h \]
Setting the volume of the cube equal to the volume of the displaced water gives:
\[ 1728 = 576 \times h \]

**Hint:** This equation relates the volume of the cube to the rise in water level.

### Step 4: Solve for the Height \( h \)
To find \( h \), rearrange the equation:
\[ h = \frac{1728}{576} \]
Calculating this gives:
\[ h = 3 \text{ inches} \]

**Hint:** When dividing, simplify the numbers if possible to make calculations easier.

### Final Answer
The water level in the tank rises by **3 inches** when the cube is completely submerged.