Solve the problem.
-A weight suspended from a spring is vibrating vertically with up being the positive direction. The function
$$f ( t ) = 10 \sin \left( \frac { 3 \pi t } { 4 } - \frac { \pi } { 4 } \right)$$ represents the distance in centimeters of the weight from its rest position as a function of time t, where t is measured in seconds. Find the smallest positive value of t for which the displacement of the weight above its rest position is 5 cm.
Round answer to three decimal places, if necessary.
A) 1.586
B) 2.293
C) 0.556
D) 0.222

Question

Quizplus · Accepted Answer

To find the smallest positive value of $t$ for which the displacement is 5 cm, set the function equal to 5 and solve for $t$:$$10 \sin \left( \frac { 3 \pi t } { 4 } - \frac { \pi } { 4 } \right) = 5$$Divide both sides by 10:$$\sin \left( \frac { 3 \pi t } { 4 } - \frac { \pi } { 4 } \right) = 0.5$$Using the inverse sine function and considering the periodic nature of the sine function, we find the angle in the first quadrant that corresponds to a sine value of 0.5, which is $\frac{\pi}{6}$. Thus, we set the argument of the sine function equal to $\frac{\pi}{6}$ and solve for $t$:$$\frac { 3 \pi t } { 4 } - \frac { \pi } { 4 } = \frac{\pi}{6}$$Solving this equation for $t$ gives $t = 0.556$ seconds, after simplifying and rounding to three decimal places.

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