For each natural number k,let C_k denote the | vedclass

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(B) The particle starts at $(1, 0)$ on circle $C_1$. On each circle $C_k$,the particle travels an arc length of $k$ cm. The angle subtended by this ar

Vedclass · Accepted Answer

$7$

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(B) The particle starts at $(1, 0)$ on circle $C_1$. On each circle $C_k$,the particle travels an arc length of $k$ cm. The angle subtended by this arc at the center is $	heta_k = \frac{	ext{arc length}}{	ext{radius}} = \frac{k}{k} = 1$ radian. After completing the motion on $C_k$,the particle moves radially to $C_{k+1}$. Let $	heta_n$ be the total angular position after completing the motion on circle $C_n$. The total angle after $n$ circles is $\sum_{k=1}^{n} 1 = n$ radians. The particle crosses the positive $x$-axis when the total angle is a multiple of $2\pi$. We need the smallest integer $n$ such that $n = 2\pi m$ for some integer $m$. Since $\pi \approx 3.14159$,$2\pi \approx 6.28$. For $n=6$,$6 For $n=7$,$7 > 6.28$. However,the particle moves radially between circles. The angular position remains constant during the radial shift. After completing $C_6$,the total angle is $6$ radians. After completing $C_7$,the total angle is $7$ radians. Since $6 Given the standard interpretation of this problem,the particle crosses the positive $x$-axis for the first time on circle $C_7$.

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