Solve the problem.
-The positions of two particles on the s-axis are $s _ { 1 } = \sin t$ and $s _ { 2 } = \sin \left( t + \frac { \pi } { 4 } \right)$, with $s _ { 1 }$ and $s _ { 2 }$ in meters and $t$ in seconds.
At what time(s) in the interval $0 \leq t \leq 2 \pi$ do the particles meet?
A) $t = \pi \sec$ and $t = 2 \pi \mathrm { sec }$
B) $t = \frac { 3 } { 8 } \pi \sec$ and $t = \frac { 11 } { 8 } \pi \sec$
C) $t = \frac { 3 } { 4 } \pi \sec$ and $t = \frac { 11 } { 4 } \pi \sec$
D) $t = \frac { 1 } { 2 } \pi \sec$ and $t = \frac { 3 } { 2 } \pi \sec$

Question

Quizplus · Accepted Answer

To find when the particles meet, we set $s_1 = s_2$, which gives $\sin t = \sin(t + \frac{\pi}{4})$. This equation is satisfied when $t + \frac{\pi}{4} = t + 2n\pi$ or $t + \frac{\pi}{4} = \pi - t + 2n\pi$, where $n$ is an integer. Solving these for $t$ within the given interval $0 \leq t \leq 2\pi$ leads to $t = \frac{3\pi}{8}$ and $t = \frac{11\pi}{8}$, which correspond to choice B.

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