AMC 8 Daily Practice Round 9

Complete problem set with solutions and individual problem pages

Problem 13 Easy

As shown in the figure, the cross-sections of 7 cylindrical chopsticks are all circles with radius r. What is the total length of the string needed to wrap around all 7 chopsticks?

  • A.

    12r + 2\pi r

  • B.

    16r + 2\pi r

  • C.

    6r + 12\pi r

  • D.

    18r

  • E.

    18 \pi r

Answer:A

As shown in the figure, let M and N be the centers of the two circles, and let BC be their external common tangent with A as the point of tangency.

Since BC = MN = 2r, the same reasoning applies to each pair of adjacent circles, forming rectangles and circular sectors.

Since there are 6 such tangents, their total length is:   6BC = 6 \times 2r = 12r

The central angle corresponding to each arc is:   6 \angle AMB = 360^\circ \quad \Rightarrow \quad \angle AMB = 60^\circ

Thus, the arc length of each circular segment is:   AB = \frac{60^\circ}{360^\circ} \times 2\pi r = \frac{1}{3} \pi r

The total string length consists of the 6 straight segments and 6 arc segments:   6BC + 6AB = 12r + 2\pi r

The answer is \text{A}.

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