Solve the problem.
-Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's positios $\mathrm { t }$.
$$v = \frac { 8 } { \pi } \sin \frac { 4 t } { \pi } , s \left( \pi ^ { 2 } \right) = 2$$
A) $\mathrm { s } = - 2 \cos \frac { 4 \mathrm { t } } { \pi } + 3.3073$
B) $s = 2 \cos \frac { 4 t } { \pi } + 4$
C) $s = - 2 \cos \frac { 4 t } { \pi } + 4$
D) $s = - 2 \cos \frac { 4 t } { \pi } + 8.2134$

Question

Quizplus · Accepted Answer

We can use integration to solve the problem. 
 $$v = \frac { ds } { dt }$$
Thus, we have 
$$ds = \frac { 8 } { \pi } \sin \frac { 4 t } { \pi } dt$$
Integrating both sides with respect to $t$, we have
$$s = \frac { 8 } { \pi } \cdot \frac { - \cos \frac { 4 t } { \pi } } { 4 } + C$$
where $C$ is the constant of integration.
Using the given initial position $s(\pi^2) = 2$, we have 
$$2 = \frac { 8 } { \pi } \cdot \frac { - \cos \frac { 4 \pi ^ 2 } { \pi } } { 4 } + C$$
Simplifying, we get
$$C = -2 + 2\cos(4\pi) = -2$$
Therefore, the position function is 
$$s = \frac { 8 } { \pi } \cdot \frac { - \cos \frac { 4 t } { \pi } } { 4 } - 2= -2\cos\frac{4t}{\pi}+4$$ 
Thus, the answer is (C).

Solve the Problem t\mathrm { t }t v=8πsin⁡4tπ,s(π2)=2v = \frac { 8 } { \pi } \sin \frac { 4 t } { \pi } , s \left( \pi ^ { 2 } \right) = 2v=π8​sinπ4t​,s(π2)=2

Solve the Problem $\mathrm { t }$ $v = \frac { 8 } { \pi } \sin \frac { 4 t } { \pi } , s \left( \pi ^ { 2 } \right) = 2$