7. At time $t \geq 0$, a particle moving in the $x y$-plane has velocity vector given by $v(t)=\left\langle 4 e^{-t}, \sin (1+\sqrt{t})\right\rangle$. What is the total distance the particle travels between $t=1$ and $t=3$ ?
(A) 1.861
(B) 1.983
(C) 2.236
(D) 4.851

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the total distance traveled by the particle, we need to integrate the magnitude of the velocity vector from $t=1$ to $t=3$. The velocity vector is given by $v(t)=\left\langle 4 e^{-t}, \sin (1+\sqrt{t})\right\rangle$. ‖ ‖ step 2 ⋮ The magnitude of the velocity vector is $\sqrt{(4 e^{-t})^2 + (\sin (1+\sqrt{t}))^2}$. ‖ ‖ step 3 ⋮ We need to integrate this magnitude from $t=1$ to $t=3$: $\int_{1}^{3} \sqrt{(4 e^{-t})^2 + (\sin (1+\sqrt{t}))^2} \, dt$. ‖ ‖ step 4 ⋮ Using the Asksia-LL calculator result, we have: $\int_{1}^{3} \sqrt{16 e^{-2 t} + \sin^2(1 + \sqrt{t})} \, dt = 1.8606$. ‖ ∻Answer∻ ⚹ 1.861 ⚹ ∻Key Concept∻ ⚹Total distance traveled by a particle⚹ ∻Explanation∻ ⚹The total distance traveled by a particle is found by integrating the magnitude of its velocity vector over the given time interval.⚹◊What is the formula for calculating total distance traveled by a particle given its velocity vector?⍭ Generate me a similar question◊

Sia