Find the circumference of the circle whose area is 16 times the area of the circle with diameter 1.4 m

Answer

To find the circumference of the circle whose area is 16 times the area of the circle with a diameter of 1.4 m, we can follow these steps:

### Step 1: Calculate the area of the smaller circle.
The formula for the area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
The diameter \( d_2 \) of the smaller circle is 1.4 m, so the radius \( r_2 \) is:
\[ r_2 = \frac{d_2}{2} = \frac{1.4}{2} = 0.7 \text{ m} \]
Now, we can calculate the area \( A_2 \):
\[ A_2 = \pi (0.7)^2 = \pi (0.49) = 0.49\pi \text{ m}^2 \]

### Step 2: Calculate the area of the larger circle.
We know that the area of the larger circle \( A_1 \) is 16 times the area of the smaller circle:
\[ A_1 = 16 \times A_2 = 16 \times 0.49\pi = 7.84\pi \text{ m}^2 \]

### Step 3: Relate the area of the larger circle to its diameter.
Using the area formula again for the larger circle, we have:
\[ A_1 = \pi r_1^2 \]
Setting the two expressions for \( A_1 \) equal gives:
\[ \pi r_1^2 = 7.84\pi \]
Dividing both sides by \( \pi \):
\[ r_1^2 = 7.84 \]
Taking the square root:
\[ r_1 = \sqrt{7.84} = 2.8 \text{ m} \]

### Step 4: Calculate the diameter of the larger circle.
The diameter \( d_1 \) is twice the radius:
\[ d_1 = 2r_1 = 2 \times 2.8 = 5.6 \text{ m} \]

### Step 5: Calculate the circumference of the larger circle.
The circumference \( C \) of a circle is given by:
\[ C = \pi d \]
Substituting the diameter \( d_1 \):
\[ C = \pi \times 5.6 \]
Using \( \pi \approx \frac{22}{7} \):
\[ C = \frac{22}{7} \times 5.6 = \frac{22 \times 5.6}{7} = \frac{123.2}{7} \approx 17.6 \text{ m} \]

### Final Answer:
The circumference of the larger circle is approximately \( 17.6 \text{ m} \).

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