The cost of turfing a triangular field at the rate of Rs. 45 per 100 `m^(2)` is Rs. 900. If double the base of the triangle is 5 times the height, then the height is :

Options

A. 50 m
B. 45 m
C. 60 m
D. 40 m

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Calculate the Area of the Triangular Field
The cost of turfing is given as Rs. 45 per 100 m², and the total cost is Rs. 900.

To find the area that can be turfed for Rs. 900, we can set up the following equation:

\[ \text{Area} = \left( \frac{900}{45} \right) \times 100 \]

Calculating this gives:

\[ \text{Area} = 20 \times 100 = 2000 \, m² \]

### Step 2: Set Up the Relationship Between Base and Height
We are given that double the base of the triangle is equal to five times the height.

Let the base be \( b \) and the height be \( h \). According to the problem:

\[ 2b = 5h \]

From this, we can express \( b \) in terms of \( h \):

\[ b = \frac{5h}{2} \]

### Step 3: Use the Area Formula for 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 area we found (2000 m²) and the expression for \( b \):

\[ 2000 = \frac{1}{2} \times b \times h \]

Substituting \( b = \frac{5h}{2} \) into the area formula:

\[ 2000 = \frac{1}{2} \times \left(\frac{5h}{2}\right) \times h \]

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

\[ 2000 = \frac{5h^2}{4} \]

To eliminate the fraction, multiply both sides by 4:

\[ 8000 = 5h^2 \]

### Step 5: Solve for \( h^2 \)
Now, divide both sides by 5:

\[ h^2 = \frac{8000}{5} = 1600 \]

### Step 6: Find the Height \( h \)
Taking the square root of both sides gives:

\[ h = \sqrt{1600} = 40 \, m \]

### Conclusion
Thus, the height of the triangular field is:

\[ \text{Height} = 40 \, m \]