The base of a triangular field is three times its altitude. If the cost of sowing the field at Rs. 58 per hectare is Rs. 783, Find its base and height.

Explanation

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

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

### 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 values we have:
\[ A = \frac{1}{2} \times (3x) \times x = \frac{3x^2}{2} \]

### Step 3: Determine the Cost of Sowing
The total cost of sowing the field is given as Rs. 783, and the cost per hectare is Rs. 58. To find the area in hectares, we can use the formula:
\[ \text{Area in hectares} = \frac{\text{Total cost}}{\text{Cost per hectare}} \]
Substituting the values:
\[ \text{Area in hectares} = \frac{783}{58} \]
Calculating this gives:
\[ \text{Area in hectares} = 13.5 \text{ hectares} \]

### Step 4: Convert Area to Square Meters
Since 1 hectare = \( 10^4 \) square meters, we convert the area:
\[ \text{Area in square meters} = 13.5 \times 10^4 = 135000 \text{ m}^2 \]

### Step 5: Set Up the Equation
Now, we set the area calculated using the altitude and base equal to the area calculated from the cost:
\[ \frac{3x^2}{2} = 135000 \]

### Step 6: Solve for \( x \)
To solve for \( x \), we first multiply both sides by 2:
\[ 3x^2 = 270000 \]
Now, divide by 3:
\[ x^2 = 90000 \]
Taking the square root of both sides gives:
\[ x = 300 \text{ meters} \]

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

### Final Answer
Thus, the altitude of the triangular field is \( 300 \) meters and the base is \( 900 \) meters.

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