Answer:
L = 8.831 t
Step-by-step explanation:
You have the following vector:
[tex]r(t)=(4+5t,8+2t,2-7t)[/tex]
To find the arc length you use the following formula:
[tex]L=\int_a^b \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2+(\frac{dz}{dt})^2}dt[/tex] (1)
where dx/dt, dy/dt and dz/dt are the components of the vector dr/dt.
Then, you calculate dr/dt:
[tex]\frac{d\vec{r}}{dt}=(5,2,-7)[/tex]
Next you replace in the integral of the equation (1):
[tex]L=\int_o^t\sqrt{(5)^2+(2)^2+(-7)^2}=8.831t|_o^t=8.831t[/tex]
hence, the arc length is L = 8.831t
