The outer circumference of a circular race-track is 528 metre. The track is everywhere 14 metre wide. Cost of levelling the track at the rate of Rs. 10 per sq. metre is :

Options

A. Rs. 77660
B. Rs. 66760
C. Rs. 76760
D. Rs. 67760

Answer

To solve the problem step by step, we need to find the cost of levelling the circular race-track given its outer circumference and width.

### Step 1: Find the radius of the outer circle
We know the outer circumference \( C \) of the circular race-track is given by the formula:
\[ C = 2\pi R \]
where \( R \) is the radius of the outer circle.

Given:
\[ C = 528 \text{ metres} \]

Substituting the value of \( C \):
\[ 528 = 2 \times \frac{22}{7} \times R \]

### Step 2: Solve for \( R \)
To find \( R \), rearranging the equation gives:
\[ R = \frac{528 \times 7}{2 \times 22} \]

Calculating the right-hand side:
\[ R = \frac{528 \times 7}{44} = \frac{3696}{44} = 84 \text{ metres} \]

### Step 3: Find the radius of the inner circle
The width of the track is given as 14 metres. Therefore, the radius of the inner circle \( r \) is:
\[ r = R - \text{width} = 84 - 14 = 70 \text{ metres} \]

### Step 4: Calculate the area of the track
The area of the track is the difference between the area of the outer circle and the inner circle. The area \( A \) of a circle is given by:
\[ A = \pi R^2 \]

Calculating the area of the outer circle:
\[ A_{\text{outer}} = \pi R^2 = \frac{22}{7} \times (84)^2 \]
Calculating \( (84)^2 \):
\[ (84)^2 = 7056 \]
Thus,
\[ A_{\text{outer}} = \frac{22}{7} \times 7056 = 22 \times 1008 = 22176 \text{ sq. metres} \]

Calculating the area of the inner circle:
\[ A_{\text{inner}} = \pi r^2 = \frac{22}{7} \times (70)^2 \]
Calculating \( (70)^2 \):
\[ (70)^2 = 4900 \]
Thus,
\[ A_{\text{inner}} = \frac{22}{7} \times 4900 = 22 \times 700 = 15400 \text{ sq. metres} \]

### Step 5: Calculate the area of the track
Now, the area of the track \( A_{\text{track}} \) is:
\[ A_{\text{track}} = A_{\text{outer}} - A_{\text{inner}} = 22176 - 15400 = 6786 \text{ sq. metres} \]

### Step 6: Calculate the cost of levelling the track
The cost of levelling the track is given by:
\[ \text{Cost} = \text{Area} \times \text{Rate} \]
where the rate is Rs. 10 per sq. metre.

Thus,
\[ \text{Cost} = 6786 \times 10 = 67860 \text{ Rs.} \]

### Final Answer
The cost of levelling the track is Rs. 67860.

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