Length of first park is 24 m and ratio between area to perimeter of this park is 36:7. Breadth of second park is 14 m and ratio between area to perimeter of this park is 63 : 16. If total cost of fencing first park is Rs. 2100, then find the cost of fencing second park?

Options

A. A)1200 Rs.
B. B)1600 Rs.
C. C)1000 Rs.
D. D)1800 Rs.

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Find the breadth of the first park.
We know:
- Length of the first park (L1) = 24 m
- Ratio of area to perimeter = 36:7

The area (A1) of the park can be expressed as:
\[ A1 = L1 \times B1 \]
Where B1 is the breadth of the first park.

The perimeter (P1) of the park is given by:
\[ P1 = 2 \times (L1 + B1) \]

From the ratio of area to perimeter, we have:
\[ \frac{A1}{P1} = \frac{36}{7} \]

Substituting the expressions for area and perimeter, we get:
\[ \frac{L1 \times B1}{2 \times (L1 + B1)} = \frac{36}{7} \]

Substituting L1 = 24 m:
\[ \frac{24 \times B1}{2 \times (24 + B1)} = \frac{36}{7} \]

### Step 2: Cross-multiply to solve for B1.
Cross-multiplying gives:
\[ 7 \times (24 \times B1) = 36 \times (2 \times (24 + B1)) \]
\[ 168B1 = 72(24 + B1) \]
\[ 168B1 = 1728 + 72B1 \]

### Step 3: Rearranging the equation.
Rearranging gives:
\[ 168B1 - 72B1 = 1728 \]
\[ 96B1 = 1728 \]

### Step 4: Solve for B1.
Dividing both sides by 96:
\[ B1 = \frac{1728}{96} = 18 \text{ m} \]

### Step 5: Calculate the perimeter of the first park.
Using the values of L1 and B1:
\[ P1 = 2 \times (L1 + B1) = 2 \times (24 + 18) = 2 \times 42 = 84 \text{ m} \]

### Step 6: Find the cost per meter of fencing.
Total cost of fencing the first park is Rs. 2100, so:
\[ \text{Cost per meter} = \frac{2100}{84} = 25 \text{ Rs/m} \]

### Step 7: Find the length of the second park.
For the second park:
- Breadth (B2) = 14 m
- Ratio of area to perimeter = 63:16

Let the length of the second park be L2. The area (A2) and perimeter (P2) can be expressed as:
\[ A2 = B2 \times L2 = 14 \times L2 \]
\[ P2 = 2 \times (B2 + L2) = 2 \times (14 + L2) \]

From the ratio:
\[ \frac{A2}{P2} = \frac{63}{16} \]

Substituting the expressions:
\[ \frac{14 \times L2}{2 \times (14 + L2)} = \frac{63}{16} \]

### Step 8: Cross-multiply to solve for L2.
Cross-multiplying gives:
\[ 16 \times (14L2) = 63 \times (2 \times (14 + L2)) \]
\[ 224L2 = 126(14 + L2) \]
\[ 224L2 = 1764 + 126L2 \]

### Step 9: Rearranging the equation.
Rearranging gives:
\[ 224L2 - 126L2 = 1764 \]
\[ 98L2 = 1764 \]

### Step 10: Solve for L2.
Dividing both sides by 98:
\[ L2 = \frac{1764}{98} = 18 \text{ m} \]

### Step 11: Calculate the perimeter of the second park.
Using the values of B2 and L2:
\[ P2 = 2 \times (B2 + L2) = 2 \times (14 + 18) = 2 \times 32 = 64 \text{ m} \]

### Step 12: Calculate the total cost of fencing the second park.
Using the cost per meter:
\[ \text{Total cost} = P2 \times \text{Cost per meter} = 64 \times 25 = 1600 \text{ Rs} \]

### Final Answer:
The cost of fencing the second park is Rs. 1600.

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