Answer:
About 141.4 cm
Explanation:
You want the length of a band that will hold 16 10-cm pipes tightly together.
Circle packing
The smallest area that will hold a given number of circles will depend on its shape. The second attachment shows the packing of 16 circles into a circular space. The gap between two of the circles in the outer ring means the polygon joining their centers will have a perimeter slightly more than 11 times a pipe diameter. We conclude the length of the metal band would be perhaps (11.2 + π) times the diameter of the pipe.
Using the packing shown in the first attachment, the length of the metal band is easily shown to be (11 +π) times the pipe diameter. This is shorter than the band required in the second attachment, so is likely the shortest possible.
Band length
The length of the metal band holding the pipes will be ...
(11 +π)(10 cm) ≈ 141.4 cm
The band is 141.4 cm long.
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Additional comment
The straight segments of the band will be parallel to and the same length as the polygon sides shown in the first attachment. The curved segments will have a total length of one pipe circumference. This is how we can tell that the band length is (11+π) times the diameter of the pipe.