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

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

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the total distance the particle travels, we need to integrate the magnitude of the velocity vector over the given time interval. ‖ ‖ step 2 ⋮ The magnitude of the velocity vector $v(t)$ is given by $\sqrt{(4e^{-t})^2 + (\sin(1+\sqrt{t}))^2}$. ‖ ‖ step 3 ⋮ We integrate this magnitude from $t=1$ to $t=3$ to find the total distance traveled. ‖ ‖ step 4 ⋮ The asksia-ll calculator has already computed this integral: $\int_{1}^{3} \sqrt{16 e^{-2t} + \sin^2(1 + \sqrt{t})} \, dt = 1.8606$. ‖ ∻Answer∻ ⚹ (A) 1.861 ⚹ ∻Key Concept∻ ⚹Integration of the magnitude of velocity⚹ ∻Explanation∻ ⚹The total distance traveled by a particle is the integral of the magnitude of its velocity vector over the time interval. The asksia-ll calculator has provided the exact value of this integral for the given time interval.⚹◊What is the formula to calculate the total distance traveled by a particle given its velocity vector in the xy-plane?⍭ Generate me a similar question◊

Sia