The base of a triangular field is three time its height. If the cost of cultivating the field at Rs. 26.38 per hectare is Rs. 356.13, find the base and height of the field.

Answer

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

### Step 1: Calculate the area of the triangular field.
The area of the field can be calculated using the formula:
\[ \text{Area} = \frac{\text{Total Cost}}{\text{Cost per hectare}} \]
Given:
- Total Cost = Rs. 356.13
- Cost per hectare = Rs. 26.38

Calculating the area:
\[ \text{Area} = \frac{356.13}{26.38} \approx 13.5 \text{ hectares} \]

### Step 2: Convert hectares to square meters.
Since 1 hectare = 10,000 square meters, we convert the area:
\[ \text{Area} = 13.5 \times 10,000 = 135,000 \text{ square meters} \]

### Step 3: Set up the relationship between base and height.
Let the height of the triangular field be \( x \) meters. According to the problem, the base \( b \) is three times the height:
\[ b = 3x \]

### Step 4: Use the area formula for a triangle.
The area \( A \) of a triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values we have:
\[ 135,000 = \frac{1}{2} \times (3x) \times x \]
This simplifies to:
\[ 135,000 = \frac{3x^2}{2} \]

### Step 5: Solve for \( x^2 \).
To eliminate the fraction, multiply both sides by 2:
\[ 270,000 = 3x^2 \]
Now, divide both sides by 3:
\[ x^2 = \frac{270,000}{3} = 90,000 \]

### Step 6: Find the value of \( x \).
Taking the square root of both sides:
\[ x = \sqrt{90,000} = 300 \text{ meters} \]

### Step 7: Calculate the base.
Now that we have the height, we can find the base:
\[ b = 3x = 3 \times 300 = 900 \text{ meters} \]

### Final Answer:
- Height = 300 meters
- Base = 900 meters

---