A cuboidal water tank has 216 litres of water. It depth is `(1)/(3)` of its length and breadth is `(1)/(2)` of `(1)/(3)` of the difference of length and deoth. The length of the tank is

Options

A. 72 dm
B. 18 dm
C. 6 dm
D. 2 dm

Answer

To find the length of the cuboidal water tank, we can follow these steps:

### Step 1: Define the Variables
Let the length of the tank be \( L \), the depth (height) be \( H \), and the breadth be \( B \).

### Step 2: Set Up the Relationships
According to the problem:
- The depth \( H \) is \( \frac{1}{3} \) of the length \( L \):
\[ H = \frac{1}{3}L \]
- The breadth \( B \) is \( \frac{1}{2} \) of \( \frac{1}{3} \) of the difference of length and depth:
\[ B = \frac{1}{2} \cdot \frac{1}{3} (L - H) \]
Substituting \( H \) into the equation:
\[ B = \frac{1}{2} \cdot \frac{1}{3} \left(L - \frac{1}{3}L\right) = \frac{1}{2} \cdot \frac{1}{3} \left(\frac{2}{3}L\right) = \frac{1}{2} \cdot \frac{2}{9}L = \frac{1}{9}L \]

### Step 3: Calculate the Volume
The volume \( V \) of the cuboidal tank can be expressed as:
\[ V = L \times B \times H \]
Substituting the expressions for \( B \) and \( H \):
\[ V = L \times \frac{1}{9}L \times \frac{1}{3}L = \frac{1}{27}L^3 \]

### Step 4: Convert Volume to Cubic Meters
The volume given is \( 216 \) liters. To convert liters to cubic meters:
\[ V = \frac{216}{1000} = 0.216 \text{ m}^3 \]

### Step 5: Set Up the Equation
Now we can set the volume equation equal to the converted volume:
\[ \frac{1}{27}L^3 = 0.216 \]

### Step 6: Solve for \( L^3 \)
Multiplying both sides by \( 27 \):
\[ L^3 = 0.216 \times 27 \]
Calculating the right side:
\[ L^3 = 5.832 \]

### Step 7: Find \( L \)
Taking the cube root of both sides:
\[ L = \sqrt[3]{5.832} \approx 1.8 \text{ m} \]

### Step 8: Convert to Decimeters
Since the answer is required in decimeters, we convert meters to decimeters:
\[ L = 1.8 \text{ m} \times 10 = 18 \text{ dm} \]

### Final Answer
The length of the tank is \( 18 \) decimeters.

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