Length and breadth of a rectangular park are `(3x^(2) + 2x)` m and `(2x^(3) – 3)` m respectively. Find the area of the park, when x = 3.

Options

A. `1924 m^(2)`
B. `1492 m^(2)`
C. `1881 m^(2)`
D. `1683 m^(2)`

Answer

To find the area of the rectangular park given the length and breadth as polynomials, we will follow these steps:

### Step 1: Define the Length and Breadth
The length \( L \) of the park is given by the polynomial:
\[ L = 3x^2 + 2x \quad \text{(in meters)} \]
The breadth \( B \) of the park is given by the polynomial:
\[ B = 2x^3 - 3 \quad \text{(in meters)} \]

### Step 2: Calculate the Area of the Park
The area \( A \) of a rectangle is calculated using the formula:
\[ A = L \times B \]
Substituting the expressions for length and breadth, we have:
\[ A = (3x^2 + 2x)(2x^3 - 3) \]

### Step 3: Substitute \( x = 3 \)
Now, we need to find the area when \( x = 3 \):
\[ L = 3(3^2) + 2(3) = 3(9) + 6 = 27 + 6 = 33 \quad \text{(meters)} \]
\[ B = 2(3^3) - 3 = 2(27) - 3 = 54 - 3 = 51 \quad \text{(meters)} \]

### Step 4: Calculate the Area
Now substitute the values of length and breadth into the area formula:
\[ A = 33 \times 51 \]

### Step 5: Perform the Multiplication
Now we calculate:
\[ 33 \times 51 = 1683 \quad \text{(square meters)} \]

### Final Answer
Thus, the area of the park is:
\[ \text{Area} = 1683 \, \text{m}^2 \]

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