A cube of metal , whose edge is 10 cm, is wholly immersed in water contained in cylindrical tube whose diameter is 20 cm, by how much will the water level rise in the tube ?

Options

A. 3.3 cm
B. `6""(3)/(11) cm `
C. `3""(2)/(11) cm `
D. none of these

Answer

To find out by how much the water level will rise in the cylindrical tube when a cube of metal is immersed in it, we can follow these steps:

### Step 1: Calculate the volume of the cube.
The formula for the volume \( V \) of a cube with edge length \( a \) is given by:
\[ V = a^3 \]
Given that the edge of the cube is 10 cm:
\[ V = 10^3 = 1000 \text{ cm}^3 \]

### Step 2: Determine the radius of the cylindrical tube.
The diameter of the cylindrical tube is given as 20 cm. The radius \( r \) is half of the diameter:
\[ r = \frac{20}{2} = 10 \text{ cm} \]

### Step 3: Write the formula for the volume of the cylindrical tube.
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( h \) is the height of the water level rise that we need to find.

### Step 4: Set the volume of the cube equal to the volume of the displaced water in the cylindrical tube.
Since the cube is wholly immersed in the water, the volume of the cube will equal the volume of the water displaced:
\[ 1000 = \pi (10^2) h \]
Substituting \( r = 10 \):
\[ 1000 = \pi (100) h \]

### Step 5: Solve for the height \( h \).
Rearranging the equation gives:
\[ h = \frac{1000}{\pi \times 100} \]
Simplifying this:
\[ h = \frac{1000}{100\pi} = \frac{10}{\pi} \]

### Step 6: Substitute the value of \( \pi \) to find \( h \).
Using \( \pi \approx \frac{22}{7} \):
\[ h = \frac{10}{\frac{22}{7}} = 10 \times \frac{7}{22} = \frac{70}{22} = \frac{35}{11} \text{ cm} \]

### Step 7: Convert to mixed fraction (if necessary).
To convert \( \frac{35}{11} \) into a mixed fraction:
\[ 35 \div 11 = 3 \quad \text{(remainder 2)} \]
Thus, \( \frac{35}{11} = 3 \frac{2}{11} \text{ cm} \).

### Final Answer:
The water level will rise by \( 3 \frac{2}{11} \text{ cm} \).

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