The base of a triangular field is 2.4 times its height. If the cost of leveling the field at the rate of rupes 20 per square metre is rupes 9,600 find its base and height.

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the Area of the Triangular Field
The cost of leveling the field is given as ₹9,600 at the rate of ₹20 per square meter. We can find the area of the field using the formula:

\[ \text{Area} = \frac{\text{Total Cost}}{\text{Cost per square meter}} \]

Substituting the values:

\[ \text{Area} = \frac{9600}{20} = 480 \text{ square meters} \]

### Step 2: Set Up the Relationship Between Base and Height
We know from the problem statement that the base (b) of the triangular field is 2.4 times its height (h). Therefore, we can express the base in terms of height:

\[ b = 2.4h \]

### Step 3: Use the Area Formula for a Triangle
The area (A) of a triangle can be calculated using the formula:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

Substituting the area we found and the expression for the base:

\[ 480 = \frac{1}{2} \times (2.4h) \times h \]

### Step 4: Simplify the Equation
Now, we simplify the equation:

\[ 480 = \frac{1}{2} \times 2.4h^2 \]

Multiplying both sides by 2 to eliminate the fraction:

\[ 960 = 2.4h^2 \]

### Step 5: Solve for Height
Now, we can solve for \(h^2\):

\[ h^2 = \frac{960}{2.4} \]

Calculating the right side:

\[ h^2 = 400 \]

Taking the square root of both sides gives us:

\[ h = 20 \text{ meters} \]

### Step 6: Find the Base
Now that we have the height, we can find the base using the relationship established earlier:

\[ b = 2.4h = 2.4 \times 20 = 48 \text{ meters} \]

### Final Answer
Thus, the base and height of the triangular field are:

- Base = 48 meters
- Height = 20 meters

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