Squares of side length 1, 1, 2, 3, and 5 are arranged to form the rectangle shown below. A curve is drawn by inscribing a quarter circle in each square and joining the quarter circles in order, from shortest to longest. What is the length of the curve ?

A. $4\pi$

B. $\frac{11}{2}\pi$

C. $6\pi$

D. $8\pi$

E. $\frac{17}{2}\pi$

1. 确定每段四分之一圆弧的半径与长度公式

每个正方形中内切的曲线为圆心角为 $90^\circ$ 的四分之一圆弧。
半径为 $r$ 的四分之一圆弧的弧长为：
$l = \frac{1}{4} \times 2\pi r = \frac{1}{2}\pi r$
五个正方形的边长即对应的各圆弧半径，分别为 $r_1 = 1$，$r_2 = 1$，$r_3 = 2$，$r_4 = 3$，$r_5 = 5$。

2. 计算整条曲线的总长度

整条曲线的总长度等于这五段四分之一圆弧长度之和：
$L = \frac{1}{2}\pi (1 + 1 + 2 + 3 + 5) = \frac{1}{2}\pi \times 12 = 6\pi$
因此，正确选项为 C。