The radius of a circle is 5 m. Find the circumference of the circle whose area is 49 times the area of the given cirlce.

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript:

### Step 1: Identify the given information
We know that the radius of the given circle (small r) is 5 m.

### Step 2: Calculate the area of the given circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of the radius:
\[ A = \pi (5)^2 = 25\pi \, \text{m}^2 \]

### Step 3: Determine the area of the new circle
According to the problem, the area of the new circle is 49 times the area of the given circle:
\[ \text{Area of new circle} = 49 \times (25\pi) = 1225\pi \, \text{m}^2 \]

### Step 4: Set up the equation for the new circle's area
Let the radius of the new circle be \( R \). The area of the new circle can also be expressed as:
\[ \pi R^2 \]
Setting the two expressions for the area equal gives:
\[ \pi R^2 = 1225\pi \]

### Step 5: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ R^2 = 1225 \]

### Step 6: Solve for \( R \)
Taking the square root of both sides:
\[ R = \sqrt{1225} = 35 \, \text{m} \]

### Step 7: Calculate the circumference of the new circle
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi R \]
Substituting the value of \( R \):
\[ C = 2\pi (35) = 70\pi \]
Using \( \pi \approx \frac{22}{7} \):
\[ C = 70 \times \frac{22}{7} = 220 \, \text{m} \]

### Final Answer
The circumference of the circle whose area is 49 times the area of the given circle is **220 m**.

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