The area of a circular park is 37 times the area of a triangular field with sides 20 m, 20 m and 24 m. What is the perimeter (nearest to an integer) of the circular park?

Options

A. 151
B. 256
C. 300
D. 359

Answer

To solve the problem, we need to follow these steps:

### Step 1: Calculate the area of the triangular field.
Given the sides of the triangle are 20 m, 20 m, and 24 m, we can use Heron's formula to find the area.

1. Calculate the semi-perimeter (s):
\[ s = \frac{a + b + c}{2} = \frac{20 + 20 + 24}{2} = 32 \text{ m} \]

2. Use Heron's formula to find the area (A):
\[ A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{32(32-20)(32-20)(32-24)} \]
\[ A = \sqrt{32 \times 12 \times 12 \times 8} = \sqrt{3072} = 48 \text{ m}^2 \]

### Step 2: Find the area of the circular park.
According to the problem, the area of the circular park is 37 times the area of the triangular field.

\[ \text{Area of circular park} = 37 \times 48 = 1776 \text{ m}^2 \]

### Step 3: Relate the area of the circular park to its radius.
The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
Setting the area equal to the area of the circular park:
\[ \pi r^2 = 1776 \]

### Step 4: Solve for the radius (r).
Using \(\pi \approx \frac{22}{7}\):
\[ \frac{22}{7} r^2 = 1776 \]
\[ r^2 = \frac{1776 \times 7}{22} = 564 \]
\[ r = \sqrt{564} \approx 23.75 \text{ m} \]

### Step 5: Calculate the perimeter of the circular park.
The perimeter (circumference) of a circle is given by:
\[ \text{Perimeter} = 2\pi r \]
Substituting the value of \(r\):
\[ \text{Perimeter} = 2 \times \frac{22}{7} \times 23.75 \approx 2 \times 3.14 \times 23.75 \approx 149.3 \text{ m} \]

### Step 6: Round to the nearest integer.
The perimeter of the circular park, rounded to the nearest integer, is approximately:
\[ \text{Perimeter} \approx 149 \text{ m} \]

### Final Answer:
The perimeter of the circular park is approximately **149 meters**.

---