Answer:
0.222 meters
Explanation:
Form the definition of instantly velocity:
[tex]v(t)=\frac{dx(t)}{dt}\\dx(t)=v(t)dt[/tex]
[tex]dx=(2te^{-3t})dt[/tex]
Integrating:
[tex]\int\limits^{x}_{0} {} \, dx=\int\limits^{t}_{0} {2te^{-3t}} \,dt[/tex]
Solve the integral by parts:
[tex]u=2t\\ du=2dt\\dv=e^{-3t}\\v=-\frac{1}{3}e^{-3t}\\[/tex]
[tex]x=-\frac{2}{3}te^{-3t}+\int\limits^{t}_{0} {\frac{2}{3} e^{-3t} } \, dt[/tex]
[tex]x=-\frac{2}{3}te^{-3t}-\frac{2}{9}e^{-3t}+\frac{2}{9}[/tex] (Remember to evaluate in t=0 and t=t)
Evaluating x(t) in t=3:
[tex]x(3)=-2e^{-9} -\frac{2}{9}e^{-9}+\frac{2}{9}\\x(3)=0.2219 [/tex] m; which rounded is 0.222 m.
