Length, breadth and height of a cuboidal tank are `(2x-y)m,(2x+y)m` and `(4x^(2)+y^(2))m` respectively. Find the volume of the tank.

Options

A. `(4x^(3)+12xy+y^(3))m^(3)`
B. `(4x^(4)+12x^(2)y^(2)+y^(2))m^(3)`
C. `(16x^(4)-y^(4))m^(3)`
D. `(16x^(4)+y^(4))m^(3)`

Answer

To find the volume of the cuboidal tank, we need to multiply its length, breadth, and height. The dimensions of the tank are given as follows:

- Length = \( (2x - y) \) m
- Breadth = \( (2x + y) \) m
- Height = \( (4x^2 + y^2) \) m

The formula for the volume \( V \) of a cuboid is:

\[ V = \text{Length} \times \text{Breadth} \times \text{Height} \]

### Step 1: Write the expression for the volume
Substituting the given dimensions into the volume formula:

\[ V = (2x - y)(2x + y)(4x^2 + y^2) \]

### Step 2: Simplify the product of Length and Breadth
We can simplify \( (2x - y)(2x + y) \) using the difference of squares formula:

\[ (2x - y)(2x + y) = (2x)^2 - y^2 = 4x^2 - y^2 \]

### Step 3: Substitute back into the volume expression
Now substitute this result back into the volume expression:

\[ V = (4x^2 - y^2)(4x^2 + y^2) \]

### Step 4: Simplify the product
Next, we can simplify \( (4x^2 - y^2)(4x^2 + y^2) \) again using the difference of squares formula:

\[ (4x^2 - y^2)(4x^2 + y^2) = (4x^2)^2 - (y^2)^2 = 16x^4 - y^4 \]

### Final Result
Thus, the volume of the cuboidal tank is:

\[ V = 16x^4 - y^4 \text{ cubic meters} \]

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