Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-A formula for the up-and-down motion of a weight on a spring is given by $\mathrm { S } = 2 \sin \left( \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \right) \mathrm { t }$, where $\mathrm { k }$ is the spring constant, $\mathrm { m }$ is the mass, and $\mathrm { t }$ is the time. Solve the equation for $\mathrm { t }$.

Question

Quizplus · Accepted Answer

To solve for $t$, we isolate $t$ in the given equation $S = 2 \sin \left( \frac { \sqrt { k } } { m } \right) t$. First, divide both sides by $2$ to get $\frac{S}{2} = \sin \left( \frac { \sqrt { k } } { m } \right) t$. Then, apply the inverse sine ($\arcsin$) to both sides to isolate the term involving $t$, resulting in $\arcsin\left(\frac{S}{2}\right) = \left( \frac { \sqrt { k } } { m } \right) t$. Finally, solve for $t$ by multiplying both sides by the reciprocal of $\frac { \sqrt { k } } { m }$, which gives $t = \frac { m } { \sqrt { k } } \arcsin \left( \frac { S } { 2 } \right)$, matching choice A.

Solve the Equation for Solutions Over the Interval [0 [0,2π)[ 0,2 \pi )[0,2π)

Solve the Equation for Solutions Over the Interval [0 $[ 0,2 \pi )$