The parallel sides of a field in the shape of a trapezium are 20 m and 41 m and the reamaining two sides are 10 m and 17 m. Find the cost of levelling the field at the rate of 30 per square meter ?

Options

A. Rs 6400
B. Rs 7320
C. Rs 7500
D. Rs 7000

Answer

To solve the problem step by step, we need to find the area of the trapezium and then calculate the cost of levelling the field based on the area.

### Step 1: Identify the dimensions of the trapezium
The parallel sides of the trapezium are given as:
- Base 1 (AB) = 20 m
- Base 2 (CD) = 41 m
The non-parallel sides are:
- Side AD = 10 m
- Side BC = 17 m

### Step 2: Drop perpendiculars from points B and C to line AD
Let’s denote the points where the perpendiculars from B and C meet line AD as points E and F, respectively.

### Step 3: Set up the equations
Since the trapezium has two parallel sides, we can denote:
- DE = x (the height of the trapezium)
- FC = 21 - x (since the total length of base CD is 41 m and DE + FC = 21 m)

### Step 4: Use the Pythagorean theorem
We will apply the Pythagorean theorem in triangles ADE and BCF.

For triangle ADE:
\[ AD^2 = AE^2 + DE^2 \]
\[ 10^2 = AE^2 + x^2 \]
\[ 100 = AE^2 + x^2 \]
(1)

For triangle BCF:
\[ BC^2 = BF^2 + CF^2 \]
\[ 17^2 = BF^2 + (21 - x)^2 \]
\[ 289 = BF^2 + (21 - x)^2 \]
(2)

### Step 5: Expand equation (2)
Expanding equation (2):
\[ 289 = BF^2 + (441 - 42x + x^2) \]
\[ BF^2 = 289 - 441 + 42x - x^2 \]
\[ BF^2 = -152 + 42x - x^2 \]
(3)

### Step 6: Set AE equal to BF
Since AE = BF, we can set equations (1) and (3) equal to each other:
\[ \sqrt{100 - x^2} = \sqrt{-152 + 42x - x^2} \]

### Step 7: Square both sides to eliminate the square roots
\[ 100 - x^2 = -152 + 42x - x^2 \]
Cancelling \( -x^2 \) from both sides:
\[ 100 = -152 + 42x \]
\[ 252 = 42x \]
\[ x = \frac{252}{42} = 6 \]

### Step 8: Calculate DE and FC
Now that we have x:
- DE = 6 m
- FC = 21 - 6 = 15 m

### Step 9: Calculate the height of the trapezium
Now we can find the area of the trapezium using the formula:
\[ \text{Area} = \frac{1}{2} \times (Base_1 + Base_2) \times \text{Height} \]
\[ \text{Area} = \frac{1}{2} \times (20 + 41) \times 6 \]
\[ \text{Area} = \frac{1}{2} \times 61 \times 6 = 183 \, \text{m}^2 \]

### Step 10: Calculate the cost of levelling the field
The cost of levelling the field at the rate of 30 per square meter:
\[ \text{Cost} = \text{Area} \times \text{Rate} \]
\[ \text{Cost} = 183 \times 30 = 5490 \, \text{rupees} \]

### Final Answer
The cost of levelling the field is **5490 rupees**.