Perimeter of a rectangular garden is `1.2xx 10^(5)m" and length "0.2xx 10^(5)m`. Find the area of the garden in standard form.

Answer

To find the area of the rectangular garden given the perimeter and length, we can follow these steps:

### Step 1: Understand the given information
We know:
- Perimeter (P) = \(1.2 \times 10^5 \, \text{m}\)
- Length (L) = \(0.2 \times 10^5 \, \text{m}\)

### Step 2: Use the formula for the perimeter of a rectangle
The formula for the perimeter of a rectangle is:
\[ P = 2 \times (L + B) \]
where \(B\) is the breadth of the rectangle.

### Step 3: Substitute the known values into the perimeter formula
Substituting the values we have:
\[ 1.2 \times 10^5 = 2 \times (0.2 \times 10^5 + B) \]

### Step 4: Simplify the equation
Divide both sides by 2:
\[ \frac{1.2 \times 10^5}{2} = 0.2 \times 10^5 + B \]
Calculating the left side:
\[ 0.6 \times 10^5 = 0.2 \times 10^5 + B \]

### Step 5: Solve for breadth (B)
Now, isolate \(B\):
\[ B = 0.6 \times 10^5 - 0.2 \times 10^5 \]
Factoring out \(10^5\):
\[ B = (0.6 - 0.2) \times 10^5 = 0.4 \times 10^5 \, \text{m} \]

### Step 6: Calculate the area of the rectangle
The area (A) of a rectangle is given by:
\[ A = L \times B \]
Substituting the values of length and breadth:
\[ A = (0.2 \times 10^5) \times (0.4 \times 10^5) \]

### Step 7: Multiply the coefficients and add the exponents
Calculating the coefficients:
\[ 0.2 \times 0.4 = 0.08 \]
For the powers of ten:
\[ 10^5 \times 10^5 = 10^{5+5} = 10^{10} \]
Thus, the area becomes:
\[ A = 0.08 \times 10^{10} \]

### Step 8: Convert to standard form
To express \(0.08\) in standard form:
\[ 0.08 = 8.0 \times 10^{-2} \]
Therefore:
\[ A = 8.0 \times 10^{-2} \times 10^{10} = 8.0 \times 10^{10 - 2} = 8.0 \times 10^8 \, \text{m}^2 \]

### Final Answer
The area of the garden in standard form is:
\[ A = 8.0 \times 10^8 \, \text{m}^2 \]