multiple choice 1 point a particle moves along the x - axis so that at any time t ≥ 0, its velocity is given…

multiple choice 1 point a particle moves along the x - axis so that at any time t ≥ 0, its velocity is given by v(t)=sin(2t). if the position of the particle at time t = π/2 is x = 4, what is the particle’s position at time t = 0? (a) -1/2 (b) 2 (c) 3 (d) 5 (e) 8

multiple choice 1 point a particle moves along the x - axis so that at any time t ≥ 0, its velocity is given by v(t)=sin(2t). if the position of the particle at time t = π/2 is x = 4, what is the particle’s position at time t = 0? (a) -1/2 (b) 2 (c) 3 (d) 5 (e) 8

Answer

Answer:

C. 3

Explanation:

Step1: Recall the relationship between velocity and position

The position function $x(t)$ is the antiderivative of the velocity function $v(t)$. So, $x(t)=\int v(t)dt=\int\sin(2t)dt$. Using the substitution $u = 2t$, $du=2dt$, we have $\int\sin(2t)dt=-\frac{1}{2}\cos(2t)+C$.

Step2: Use the given condition to find C

We know that when $t = \frac{\pi}{2}$, $x = 4$. Substitute these values into $x(t)=-\frac{1}{2}\cos(2t)+C$. $x(\frac{\pi}{2})=-\frac{1}{2}\cos(2\times\frac{\pi}{2})+C$. Since $\cos(\pi)= - 1$, then $4=-\frac{1}{2}\times(-1)+C$. $4=\frac{1}{2}+C$, so $C = 4-\frac{1}{2}=\frac{7}{2}$.

Step3: Find the position at $t = 0$

Now that $x(t)=-\frac{1}{2}\cos(2t)+\frac{7}{2}$, substitute $t = 0$. $x(0)=-\frac{1}{2}\cos(0)+\frac{7}{2}$. Since $\cos(0)=1$, then $x(0)=-\frac{1}{2}\times1+\frac{7}{2}=\frac{-1 + 7}{2}=3$.