A rectangular lawn whose length is one-half times of its breadth. The area of the law is `2/3` hectares. The length of the lawn is:

Options

A. 100 metres
B. `33 1/3` metres
C. `66 2/3` metres
D. `(100/sqrt3)` metres

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define the Variables
Let the breadth of the lawn be denoted as \( B \) and the length as \( L \). According to the problem, the length is one-half of the breadth.

\[ L = \frac{1}{2} B \]

### Step 2: Write the Area Formula
The area \( A \) of a rectangle is given by the formula:

\[ A = L \times B \]

### Step 3: Substitute the Length in the Area Formula
Substituting the expression for length \( L \) into the area formula:

\[ A = \left(\frac{1}{2} B\right) \times B = \frac{1}{2} B^2 \]

### Step 4: Set the Area Equal to Given Value
We know from the problem that the area of the lawn is \( \frac{2}{3} \) hectares. To work in square meters, we convert hectares to square meters (1 hectare = 10,000 square meters):

\[ \frac{2}{3} \text{ hectares} = \frac{2}{3} \times 10,000 = \frac{20,000}{3} \text{ square meters} \]

Now we can set the area equal to this value:

\[ \frac{1}{2} B^2 = \frac{20,000}{3} \]

### Step 5: Solve for Breadth \( B \)
To eliminate the fraction, multiply both sides by 2:

\[ B^2 = \frac{20,000 \times 2}{3} = \frac{40,000}{3} \]

Now, take the square root of both sides to find \( B \):

\[ B = \sqrt{\frac{40,000}{3}} = \frac{\sqrt{40,000}}{\sqrt{3}} = \frac{200}{\sqrt{3}} \]

### Step 6: Find the Length \( L \)
Now that we have \( B \), we can find \( L \):

\[ L = \frac{1}{2} B = \frac{1}{2} \times \frac{200}{\sqrt{3}} = \frac{100}{\sqrt{3}} \]

### Step 7: Final Answer
Thus, the length of the lawn is:

\[ L = \frac{100}{\sqrt{3}} \text{ meters} \]