2006 AMC 10B Problem 6

Below is the professionally curated solution for Problem 6 of the 2006 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 10B solutions, or check the answer key.

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Concepts:arc perimeter circumference

Difficulty rating: 1060

A region is bounded by semicircular arcs constructed on the sides of a square whose sides measure $\tfrac{2}{\pi},$ as shown. What is the perimeter of this region?

$\dfrac{4}{\pi}$

$2$

$\dfrac{8}{\pi}$

$4$

$\dfrac{16}{\pi}$

Solution:

Each side has length $\tfrac{2}{\pi},$ the diameter of a semicircular arc, so each arc has length $\tfrac12\pi\cdot\tfrac{2}{\pi}=1.$

The boundary consists of four such arcs, so the perimeter is $4\cdot1=4.$

Thus, the correct answer is D.

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