The base of a triangular field is three times its height . If the cost of cultivating the field at Rs. 367.20 per hectare is Rs. 4957.20 find its base and height.

Answer

To solve the problem, we need to find the base and height of a triangular field given that the base is three times its height. We also know the cost of cultivating the field and can use that to find the area.

### Step-by-Step Solution:

1. **Identify the relationship between base and height**:
- Let the height of the triangular field be \( h \).
- According to the problem, the base \( b \) is three times the height. Therefore, we can write:
\[ b = 3h \]

2. **Calculate the area of the triangular field**:
- The total cost of cultivating the field is given as Rs. 4957.20, and the cost per hectare is Rs. 367.20.
- To find the area in hectares, we divide the total cost by the cost per hectare:
\[ \text{Area} = \frac{\text{Total Cost}}{\text{Cost per hectare}} = \frac{4957.20}{367.20} = 13.5 \text{ hectares} \]

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

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:
\[ 135,000 = \frac{1}{2} \times b \times h \]
- Since \( b = 3h \), we can substitute for \( b \):
\[ 135,000 = \frac{1}{2} \times (3h) \times h \]
- This simplifies to:
\[ 135,000 = \frac{3}{2} h^2 \]

5. **Solve for height \( h \)**:
- Multiply both sides by 2 to eliminate the fraction:
\[ 270,000 = 3h^2 \]
- Divide both sides by 3:
\[ h^2 = 90,000 \]
- Taking the square root of both sides gives:
\[ h = \sqrt{90,000} = 300 \text{ meters} \]

6. **Calculate the base \( b \)**:
- Now that we have the height, we can find the base:
\[ b = 3h = 3 \times 300 = 900 \text{ meters} \]

### Final Answer:
- Height \( h = 300 \) meters
- Base \( b = 900 \) meters