A particle moves along the x-axis so that at time t≥0 its position is given by x(t)=2t3−9t2−60t+4. What is the total distance traveled by the particle over the time interval 0≤t≤7 ?

Answer:-175 or 175  along the x-axis in a negative directionStep-by-step explanation:The distance is the total length of the trajectory made by a moving object between two points. We need to find the total distance traveled by a particle over the time interval  [tex]t\in[0,7][/tex] , so:Let:[tex]d_o=Distance\hspace{3}traveled\hspace{3}at\hspace{3}t=0\\d_f=Distance\hspace{3}traveled\hspace{3}at\hspace{3}t=7[/tex]Using the equation provided by the problem:[tex]x(t)=2t^3-9t^2-60t+4[/tex]For t=0[tex]x(0)=2*(0)^3-9*(0)^2-60*(0)+4=0-0-0+4=4[/tex]For t=7[tex]x(7)=2*(7)^3-9*(7)^2-60*(7)+4=686-441-420+4=-171[/tex]Hence, the total distance traveled by the particle over the time interval 0≤t≤7 is:[tex]Total\hspace{3}distance\hspace{3}traveled=d_t=d_f-d_o=-171-4=-175[/tex]