Two weights (weight 1 and weight 2 ) are suspended from the ceiling by springs. At time $\mathrm{t}=0$ ( $\mathrm{t}$ in seconds), the weights are set in motion and begin bobbing up and down. Eventually, however, the oscillation of both weights dies down. The following equations describe the distance of each weight from the ceiling as a function of time:
$d_{1}=6+5 \cos (\pi t) e^{-0.1 t} \text { and } d_{2}=5+4 \cos (2 \pi t) e^{-0.3 t}$
At what time are the two weights furthest apart?
A) At $t=0$.
B) Between $t=0$ and $t=0.5$.
C) At $t=0.5$.
D) Between $t=0.5$ and $t=1$.
E) At $t=1$.

Question

Quizplus · Accepted Answer

The distance between the weights is given by $d_1 - d_2$. To find when they are furthest apart, we need to maximize this expression. The exponential decay terms $e^{-0.1t}$ and $e^{-0.3t}$ cause the oscillations to decrease over time, so the maximum separation is likely to occur early on, between $t=0$ and $t=0.5$.

Two Weights (Weight 1 and Weight 2 ) Are Suspended t=0\mathrm{t}=0t=0

Two Weights (Weight 1 and Weight 2 ) Are Suspended $\mathrm{t}=0$