The base of a triangular field is three times its heigth. If the cost of cultivating the field is Rs. 36.72 per hectares is Rs. 495.72 , find the heigth and base of the triangular field :

Options

A. 480 m, 1120 m
B. 400m, 1200 m
C. 300 m , 900 m
D. 250 m, 650 m

Answer

To solve the problem, we need to find the height and base of a triangular field given that the base is three times its height and the cost of cultivating the field is provided. Let's break down the solution step by step.

### Step 1: Define Variables
Let the height of the triangular field be \( x \) meters. Since the base is three times the height, we can express the base as:
\[ \text{Base} = 3x \text{ meters} \]

### 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:
\[ A = \frac{1}{2} \times (3x) \times x = \frac{3x^2}{2} \text{ square meters} \]

### Step 3: Convert the Total Cost to Area
We know the total cost of cultivating the field is Rs. 495.72, and the cost per hectare is Rs. 36.72. To find the area in hectares, we can use the formula:
\[ \text{Area in hectares} = \frac{\text{Total Cost}}{\text{Cost per hectare}} = \frac{495.72}{36.72} \]
Calculating this gives:
\[ \text{Area in hectares} = 13.5 \text{ hectares} \]

### Step 4: Convert Hectares to Square Meters
Since 1 hectare is equal to 10,000 square meters, we convert the area:
\[ \text{Area in square meters} = 13.5 \times 10,000 = 135,000 \text{ square meters} \]

### Step 5: Set the Areas Equal
Now we can set the area calculated from the dimensions equal to the area calculated from the cost:
\[ \frac{3x^2}{2} = 135,000 \]

### Step 6: Solve for \( x^2 \)
To eliminate the fraction, multiply both sides by 2:
\[ 3x^2 = 270,000 \]
Now, divide by 3:
\[ x^2 = 90,000 \]

### Step 7: Find \( x \)
Taking the square root of both sides gives:
\[ x = \sqrt{90,000} = 300 \text{ meters} \]

### Step 8: Calculate the Base
Now that we have the height, we can find the base:
\[ \text{Base} = 3x = 3 \times 300 = 900 \text{ meters} \]

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

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