A rectangular garden of length `(2x^(3) + 5x^(2) – 7)` m has the perimeter `(4x^(3) - 2x^(2) + 4)` m. Find the breadth of the garden.

Options

A. `(6x^(2) -9 ) `m
B. `(-6x^(2) +9) ` m
C. `(2x^(3) - 7x^(2) + 11) m `
D. `(6x^(3) +7x^(2) +9)` m

Answer

To find the breadth of the rectangular garden, we can use the formula for the perimeter of a rectangle. The perimeter \( P \) of a rectangle is given by:

\[ P = 2(\text{length} + \text{breadth}) \]

Given:
- Length \( L = 2x^3 + 5x^2 - 7 \) m
- Perimeter \( P = 4x^3 - 2x^2 + 4 \) m

### Step 1: Set up the equation for the perimeter

Using the perimeter formula, we can set up the equation:

\[ 4x^3 - 2x^2 + 4 = 2((2x^3 + 5x^2 - 7) + \text{breadth}) \]

### Step 2: Simplify the equation

First, we simplify the right side of the equation:

\[ 4x^3 - 2x^2 + 4 = 2(2x^3 + 5x^2 - 7) + 2 \cdot \text{breadth} \]

Calculating \( 2(2x^3 + 5x^2 - 7) \):

\[ = 4x^3 + 10x^2 - 14 \]

Thus, the equation becomes:

\[ 4x^3 - 2x^2 + 4 = 4x^3 + 10x^2 - 14 + 2 \cdot \text{breadth} \]

### Step 3: Rearranging the equation

Now, we can rearrange the equation to isolate the breadth:

\[ 4 = 10x^2 - 2x^2 - 14 + 2 \cdot \text{breadth} \]

This simplifies to:

\[ 4 = 8x^2 - 14 + 2 \cdot \text{breadth} \]

Adding 14 to both sides:

\[ 18 = 8x^2 + 2 \cdot \text{breadth} \]

### Step 4: Isolate the breadth

Now, isolate \( 2 \cdot \text{breadth} \):

\[ 2 \cdot \text{breadth} = 18 - 8x^2 \]

Dividing by 2 gives:

\[ \text{breadth} = \frac{18 - 8x^2}{2} \]

### Step 5: Simplify the breadth

This simplifies to:

\[ \text{breadth} = 9 - 4x^2 \]

### Final Answer

Thus, the breadth of the garden is:

\[ \text{breadth} = 9 - 4x^2 \text{ m} \]