The base of a triangular field is three times its height. If the cost of cultivating the field at Rs.1080 per hectare is Rs.14580, find its base and height.

Answer

To solve the problem step by step, we will find the base and height of the triangular field given the conditions in the question.

### Step 1: Define Variables
Let the height of the triangular field be \( h \) meters. According to the problem, the base \( b \) is three times the height. Therefore, we can express the base as:
\[ b = 3h \]

### Step 2: Calculate the Area of the Triangle
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the expressions for base and height, we get:
\[ A = \frac{1}{2} \times (3h) \times h = \frac{3}{2} h^2 \]

### Step 3: Determine the Total Area from Cost
We know the total cost of cultivating the field is Rs. 14,580, and the cost per hectare is Rs. 1,080. To find the area in hectares, we use the formula:
\[ \text{Area in hectares} = \frac{\text{Total Cost}}{\text{Cost per hectare}} = \frac{14580}{1080} = 13.5 \text{ hectares} \]
Now, converting hectares to square meters (1 hectare = 10,000 m²):
\[ \text{Area in square meters} = 13.5 \times 10000 = 135000 \text{ m}^2 \]

### Step 4: Set Up the Equation
Now we have two expressions for the area:
1. From the dimensions of the triangle: \( A = \frac{3}{2} h^2 \)
2. From the cost: \( A = 135000 \)

Setting these equal gives:
\[ \frac{3}{2} h^2 = 135000 \]

### Step 5: Solve for Height
To isolate \( h^2 \), multiply both sides by \( \frac{2}{3} \):
\[ h^2 = 135000 \times \frac{2}{3} = 90000 \]
Now, take the square root of both sides to find \( h \):
\[ h = \sqrt{90000} = 300 \text{ meters} \]

### Step 6: Calculate the Base
Now that we have the height, we can find the base using the relationship \( b = 3h \):
\[ b = 3 \times 300 = 900 \text{ meters} \]

### Conclusion
The height of the triangular field is \( 300 \) meters and the base is \( 900 \) meters.

### Summary of Results
- Height \( h = 300 \) meters
- Base \( b = 900 \) meters