To find the length of the string that is tightly tied around three circles of radius 4 cm each, we need to consider both the distances around the circles and the straight segments between them. Here's how to approach the problem:

Step 1: Understanding the Configuration

• The three circles are of equal radius $r = 4$ cm.
• Since the circles are touching each other, the centers of these circles form an equilateral triangle.

Step 2: Calculate the Side Length of the Equilateral Triangle

The distance between the centers of any two touching circles is equal to twice the radius: $\text{Side length of the equilateral triangle} = 2r = 2 \times 4 = 8 \text{ cm}$

Step 3: Calculate the Perimeter of the Triangle

The perimeter of the equilateral triangle formed by the centers of the three circles is: $\text{Perimeter of the triangle} = 3 \times 8 = 24 \text{ cm}$

Step 4: Consider the Arc Lengths

Since the string is wrapped around the three circles, it will cover three arcs that are part of each circle. The angle subtended by each arc at the center of the circle is $60^\circ$ (since the triangle is equilateral).

The arc length for one $60^\circ$ segment of the circle is given by: $\text{Arc length} = \frac{60^\circ}{360^\circ} \times 2\pi r = \frac{1}{6} \times 2\pi \times 4 = \frac{8\pi}{6} = \frac{4\pi}{3} \text{ cm}$ For three arcs: $\text{Total arc length} = 3 \times \frac{4\pi}{3} = 4\pi \text{ cm}$

Step 5: Calculate the Total Length of the String

The total length of the string is the sum of the perimeter of the triangle and the total arc length: $\text{Total string length} = \text{Perimeter of the triangle} + \text{Total arc length} = 24 \text{ cm} + 4\pi \text{ cm}$

Step 6: Numerical Value

Substituting the value of $\pi \approx 3.1416$: $\text{Total string length} \approx 24 + 4 \times 3.1416 \approx 24 + 12.5664 \approx 36.5664 \text{ cm}$

Final Answer:

The length of the string is approximately 36.57 cm.

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Do you want more details on this or have any questions?

Here are five related questions:

1. What would be the length of the string if the circles had a different radius?
2. How does the string length change if the circles are not touching but have gaps?
3. Can you find the area of the triangle formed by the centers of the circles?
4. What is the total area covered by the string, including the circles?
5. How would the solution change if there were four circles instead of three?

Tip: Understanding the relationship between circular arcs and straight-line segments is crucial in solving problems involving shapes formed by circles and tangential lines.